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Statistical Hydrodynamics (Onsager Revisited)
CHAPTER
1
Statistical Hydrodynamics (Onsager Revisited) Raoul Robert Institut Fourier CNRS, Universit6 Grenoble 1, UFR de math6matiques, BP 74, 38402 Saint Martin d'Hbres cedex, France E-mail: raoul, robert@ ujf-grenoble.fr
Contents I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Equilibrium problems: Self-organization of the turbulent flow . . . . . . . . . . . . . . . . . . . . . . . 2.1. Equilibrium states for 2D incompressible ideal flows . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Extension of the theory to other systems, the Vlasov-Poisson equation . . . . . . . . . . . . . . . 2.3. Relaxation towards the equilibrium and parametrization of the small scales . . . . . . . . . . . . . 3. Out-of-equilibrium problems: Weak solutions, shocks, and energy dissipation . . . . . . . . . . . . . . 3. I. Statistical solutions of I D inviscid Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Energy dissipation for 3D flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Last comments and acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H A N D B O O K OF M A T H E M A T I C A L FLUID DYNAMICS, V O L U M E II Edited by S.J. Friedlander and D. Serre 9 2003 Elsevier Science B.V. All rights reserved
3 4 6 25 31 37 37 44 50 50 5i
Statistical hydrodynamics (Onsager revisited)
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1. Introduction
Though it would be soundly enlightening to discuss thoroughly how statistical methods were introduced into the field of turbulence, we intend to limit ourself to only a few preliminary comments. In its crudest form the statistical approach consists in splitting the turbulent velocity field into a smoothly varying part plus a random fluctuation. This naturally yields the notion of turbulent viscosity (Reynolds, Boussinesq). From the beginning of the last century, systematic efforts were made by prominent scientists to elucidate the enigma of turbulence by statistical methods. In most cases these efforts amount to perform more or less formal calculations on random solutions of Navier-Stokes equation, solutions which are supposed to exist, provide a relevant description of turbulent flows and moreover satisfy some symmetry properties. The main success of this approach is the famous 4/5 law of Kolmogorov, Von Karm~.n and Howarth which gives a precise relationship between the energy dissipation and the third moment of the velocity increments (see Frisch [25]). More recently further progress was made on the mathematical aspect of this statistical approach when two important issues were addressed: (l) Give a precise definition of statistical solutions. (2) Prove existence results for such solutions. For these aspects we refer to [24]. Our purpose here is to expound some developments of the statistical approach which are directly inspired by the original ideas of Onsager on turbulence [46]. Onsager's starting point is a clear distinction between the two and three-dimensional cases: in two dimensions the energy of the turbulent flow is conserved, and the main phenomenon to explain is the self-organization of the flow into coherent structures (large scale eddies). On the other hand, in three dimensions, the dissipation of the energy persists in the limit of vanishing viscosity and we observe a power law energy spectrum. The distinction made by Onsager is brilliantly shown by the experiments of Van Heijst [68]. In the first part we will try to carry through Onsager's program on 2D turbulence, that is, try to extend in a rigorous way to hydrodynamics the statistical mechanics approach of Boltzmann. We will also indicate some practical consequences concerning numerical simulations. In the second part we consider the 3D case. Here also we follow Onsager's ideas: we do not start considering solutions of Navier-Stokes equation and then make the viscosity go to zero (what we may call Leray's point of view) but we suppose that the turbulent flow might be correctly described by weak solutions of Euler equation which are not regular enough to conserve the energy. Despite the fact that the Cauchy problem for such solutions is still unsolved, examples of such weak solutions with energy dissipation have been recently constructed. We begin by considering the simpler case of the one-dimensional Burgers equation which retains some of the interesting features of our problem (energy dissipation for weak solutions with shocks). Due to the discontinuities in the solution the energy will decrease. It is natural to try a statistical description of the dynamics of these shocks, that is, to describe the action of Burgers flow on a stochastic process. A first answer to this issue appears astonishingly simple: the class of Levy's process with negative jumps is preserved by Burgers equation.
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Of course, it is tempting to imagine a similar approach for Euler equation. Obviously, things are much more difficult. As a first step we propose a formula giving the expression of local energy dissipation due to the lack of regularity of weak solutions. This leads to a natural entropy condition for such solutions.
2. Equilibrium problems: Self-organization of the turbulent flow The most striking feature of 2D hydrodynamical turbulence is the emergence of a largescale organization of the flow, leading to structures usually called coherent structures (see references in [17,54]). Jupiter's Great Red Spot, a huge vortex persisting for more than three centuries in the turbulent shear between two zonal jets, is probably related to this general property [40,64]. Such hydrodynamical vortices, whose dynamics is governed by Euler equation or some quasi-geostrophic variant, occur in a wide variety of geophysical phenomena and their robustness demands a general understanding. Similarly, the galaxies themselves follow a kind of organization revealed in the Hubble classification [11]. The dynamics of Galaxies is dominated by stars under collective gravitational interaction rather than gas or hydrodynamical processes. In particular, for most stellar systems the collisions (i.e., close encounters) between stars are quite negligible and the galaxy dynamics is well modeled by the Vlasov-Poisson equation. The common remarkable feature of these structures is that they occur and persist in a strongly turbulent environment. In the case of 2D turbulence, Onsager [46] was the first to suggest that an explanation might be found in terms of statistical mechanics of Euler equation. In the case of galaxies, the natural approach, which consists in defining a statistical equilibrium for a cloud of stars, fails (see references in [17]); this is mainly due to the fact that the relaxation time associated with the "collisions" of stars widely exceeds the age of the universe. This lead astrophysicists to suggest that it is a much more efficient mixing process generated by the collisionless Vlasov-Poisson equation which drive the system towards some sort of equilibrium. In [ 17] we stress, at a physical level, the analogy between the violent relaxation of stellar systems and the mixing of vorticity yielding coherent structures in 2D turbulence. This analogy resides in the similar morphology of the Euler and Vlasov equations, and we try to explain their self-organization with the same theoretical tools. Let us now be more precise. In his pioneering paper, Onsager's argument on the possibility of negative temperature equilibrium states was based on the approximation of the continuous Euler system by a great (but finite) number of point vortices. This leads to a finite-dimensional Hamiltonian system, to which the methods of statistical mechanics can be applied (see, for example, [13,45]). Though qualitatively very enlightening, this approach reveals severe difficulties; for example, there are many different ways to approximate a continuous vorticity by a cloud of point vortices and different approximations can lead to very different statistical equilibrium states, so the thermodynamical equilibrium that we can associate to a continuous vorticity depends dramatically on arbitrary choices (this difficulty was underlined by Onsager). Thus, if we want to produce reliable quantitative predictions on the actual behavior of the flow we have to proceed differently.
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A natural way to define equilibrium states is to construct invariant Gibbs measures on the phase space. But we do not know how to construct such measures on the natural phase space L ~ (Y2) for Euler equation. Some work has been devoted to the study of Gibbs measures with formal densities given by the enstrophy and the energy [8], and also to Gibbs measures associated with the law of vorticity conservation along the trajectories of the fluid particles [ 12]. Unfortunately, all these measures are supported by "large" functional spaces so that not only the mean energy and enstrophy of these states are infinite but the phase space L ~ ( s is of null measure. So, it is only at a formal level that this makes sense. Moreover this approach fails to give any prediction on the long time dynamics corresponding to a given initial vorticity function. The most common approach to overcome these difficulties is to use a convenient finitedimensional approximation of the system, possessing an invariant Liouville measure. Then one can consider the canonical measures associated with the constants of the motion and try to perform a thermodynamic limit in the space of generalized functions when the number of degrees of freedom goes to infinity. For example, for Euler equations one can consider the N Fourier-mode approximation or the point-vortex approximation. Two difficulties arise in this approach. The first is to choose a relevant scaling to perform the limit, the second is even more fundamental: generally, the approximate system will have less constants of the motion than the continuous one, so that the long-time dynamics of that system may be very different from that of the continuous one. For more comments and references on these attempts see, for example, [23,49,54]. To overcome the difficulties evoked above, our approach is based on the following points. 9 We work on an extended phase space (the space of Young measures) on which the constants of the motion put natural constraints. 9 We construct a sequence of finite-dimensional approximations of the Euler flow (with good convergence properties such as strong L 2 uniform convergence on any finite time interval), satisfying the two tbllowing properties: (i) A Liouville theorem holds for the finite-dimensional approximations. (ii) For the family of measures given by (i), we can prove the Sanov-type large deviation estimates for empirical Young measures which are necessary to take the thermodynamic limit [41,49,52]. Of course, it is easy to satisfy the point (i) by considering the spectral approximation; but then, it is a very difficult issue to prove that the associated family of measures satisfies (ii). Our approach here is to get a Liouville theorem for a general class of approximations, including approximations on spaces of functions which are spatially localized like finite element approximants. For such approximations we are not able to prove directly the large deviation estimates (ii) but we can use them as an intermediate to construct the final approximation on the space of piecewise constant functions for which the large deviation estimates hold [41]; so that the use of the finite-element approximants appears here as an essential intermediate step in order to both insure the convergence of the approximations and keep the large deviation estimates. One might worry about the fact that our approximate dynamical system retains only the enstrophy among the infinite family of the Casimir functionals which are conserved by the continuous system (in contrast with the finite mode Hamiltonian approximation
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R. Robert
of [72,73]). Of course, it would be more satisfactory to construct approximations having in addition a large number of constants of the motion. But we think that this is not truly necessary to our microcanonical approach. Indeed, if we are interested in the long-time behavior of 2D-Euler flow, and if we believe that a statistical mechanics approach can bring some light on this issue, then we expect that we will finally have to solve some constrained variational problem: find the maximum value of some entropy functional under a set of constraints. But while we have no doubts about the set of constraints which is directly derived from the constants of the motion of the system (energy, integrals of functions of the vorticity field ... ), it is hard to guess what the relevant entropy functional is. But in our microcanonical approach the entropy is not related to the fact that many constants of the motion are (or are not) exactly conserved by the approximate flow but it is only associated to large deviation estimates for the invariant measures.
2.1. Equilibrium states for 2D incompressible ideai flows 2.1.1. 2D Euler equation. The motion of a two-dimensional incompressible inviscid fluid in a bounded domain s is governed by Euler equation, which we write in the classical velocity-vorticity formulation:
(E)
{wl + div(wu) - 0, curlu--(o, divu--0,
u.n=O
on~)s
where u(t, x) is the velocity field of the fluid, ~o = curl u the scalar vorticity, n the outward unit normal vector to ~)s Because of incompressibility we introduce the stream function
O(t, x): ~o=-/xg,,
~=0
on 0,(2.
The constants of the motion of this dynamical system are: - the energy
,fu
,T, ( w ) - ~
dx-
~
~codx;
- the integrals
F0(~o)- fa O(o)(x))dx, for any continuous function 0. These constants of the motion which are associated to the degeneracy of the (infinite-dimensional) Hamiltonian system are usually called Casimir functionals.
Statistical hydrodynamics (Onsager revisited) -
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if #2 is the ball B(0, R), we must consider also the angular momentum with respect to 0: M(~) =
'(fo
x A u(x) dx = ~
(R 2 - x 2 ) ~ ( x ) dx
)
e3.
The Cauchy problem. Yudovich's theorem [70] gives a satisfactory existence-uniqueness result for the Cauchy problem for (E): For any given initial datum co0(x) in the space L ~ ( ~ ) , there is a unique weak solution of (E); this solution co(t, x) is in L ~ ( ~ ) for all t, and furthermore belongs to the space C([0, ~ [ ; LP(ff2)) for all p, l~
The turbulent mixing process. Let us now briefly describe the mechanism of turbulent mixing which is responsible for the self-organization of the flow in Euler equation. As we have seen, Euler equation can be described as the advection of a scalar function (the vorticity) by an incompressible velocity field with which it interacts via a Poisson equation. The vorticity is not passively advected by the flow but is coupled with its motion, this coupling will be responsible for the fluctuations of the stream function which will mix the vorticity at small scale and induce a self-organization and the appearance of structures at larger scales. This process is studied at a physical level in [17]. Our concern here is to introduce an entropy functional which will give a precise content to the vague notion of turbulent disorder of the flow. Of course, to define such a functional, following Boltzmann's approach, one would have to choose a relevant measure on the phase space. It is well known that, at a formal level, Euler equation is an infinite-dimensional Hamiltonian system; but, in contrast with the finite-dimensional case, this does not imply the existence of an invariant Liouville measure on the natural phase space L ~ . Fortunately, it occurs that to define the entropy functional we do not actually need to have a Liouville measure on the infinite-dimensional phase space, we only need the existence of finite-dimensional approximations which admit invariant Liouville measures, large deviation theory will then yield the relevant entropy. This is the very root of our thermodynamical approach. Although we can find finite-dimensional approximations of Euler equation which preserves the Hamiltonian structure [72,73], this structure is broken by any kind of approximation of practical use. But tbr the needs of thermodynamics the Hamiltonian structure is not truly necessary, it is the Liouville theorem and the constants of the motion which are the key ingredients. In the case of Euler equation, it is well known that a Liouville theorem holds for the usual spectral approximation. We shall show that this is a particular case of a general property: there is a natural way to approximate Euler equation on any finite-dimensional space in such a way that the volume measure is conserved. The spectral approximation is only a particular case of that. Then the problem of defining an equilibrium statistical
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R. Robert
mechanics for (E) amounts to the study of families of measures. For an arbitrary choice of the approximating spaces the study of the asymptotic behavior of these measures seems untractable, but fortunately we can choose spaces for which the thermodynamic limit of these measures can be carried on [41,52]. 2.1.2. Finite-dimensional approximations and Liouville theorem. A classical way to construct finite-dimensional approximations of Euler equation is as follows. Let FN be an N-dimensional subspace of L ~ ( ~ ) and denote by PN the orthogonal projector from Lz(~Q) onto FN. Then we define the approximate solution coN(t) as the solution of the ordinary differential equation in FN:
(EN)
where
U N -- curl
coN + PN(U N. Vco N) --O, (.0N (0) -- PN COO
1/./"N,a n d
- - A 1 / f N - - coN, 1//,N _ 0 o n 0 F 2 .
If FN is properly chosen and w0 is regular enough, then (_,ON (t) converges towards co(t) for the strong L 2 topology, uniformly on any bounded time interval [39]. The constants of the motion of the dynamical system (EN) are: - the energy 89fs2 1//'U CONdx, - the enstrophy J~2 (CON)2 dx. Let us notice here that (EN) is a differential system with quadratic nonlinearity so that the solution always exists on a small time interval; but due to the conservation of the enstrophy the solution cannot blow up and it exists globally in time. Now, it is well known that if we take for FN a subspace generated by N eigenvectors of the operator - A (with the Dirichlet boundary condition), the volume measure on FN is conserved by (EN). This is in fact a particular case of what follows. We consider the modification of ( E N ) which consists in replacing, in the definition of ~ U the Dirichlet problem by the variational formulation:
g/NEFNand
fs27~tN.7~odx=fs2coN~odx,
for all r in FN.
For the sake of simplicity, from now on we shall also denote by (EN) this modified dynamical system. Of course, we shall suppose at least that FN is included in the Sobolev space Hr ( ~ ) , so that for any given w N, the above variational problem possesses a unique
solution 1/J u (by the Lax-Milgram theorem). One can easily check that the energy and the enstrophy are still conserved but now we have in addition THEOREM 2. I.I. The volume measure on FN is conserved by the dynamical system (EN). PROOF. FN is endowed with the L 2 scalar product. Let us write (EN) in the form coN = G N (CON), where G N(cou ) -- -- PN ( uN .TcoU) is a nonlinear transformation of FN. Then to
Statistical hydrodynamics (Onsager revisited)
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prove the theorem it suffices to show that the trace of the d e r i v a t i v e GiN(coN) vanishes. Let us compute the first variation of GN corresponding to a small variation ~coN. -
o;,
] -
-
-
By definition, we have tr(G~v (coN)) -- ~--~i(G%(coN)[ei], ei), for any orthonormal basis ei of FN. Let us denote ui the vector field associated to ei, we have:
(G'N(coN)[ei],ei)---- fs2(ui'VcoN)eidx-- fs2(uN'Vei)eidx, but since div U N "~ 0, the last term vanishes, and after integration by parts we get:
(G~N(coN)[ei], el) -- f ~ coN curl ~i " Vei dx. Let us consider now the positive definite and symmetric linear operator A defined on FN by: fs2 V7 r" X7qgdx = (ATr, qg), and take for ei an orthonormal basis of eigenvectors of A. We obviously have ei -~ )~i~i (~.i is the eigenvalue corresponding to e i ) , so that curl ~ i . V e i -- 0 and tr(G~(cox)) -- 0. I-7 Now, it remains to prove the convergence (when N --+ (x~) of the approximate solution coN (t) towards the solution co(t) of the Euler equation.
We shall take for approximating space FN the space Fh (s of the finite-element approximation of the Sobolev space H m (R2), with compact support in s (m is an integer > 5 and h a small positive parameter, see Appendix A). Then we have the following convergence result whose proof is classical. PROPOSITION 2.1.2. Let co(t) be any weak solution of (E), with co0(x) in the .space
L~(S2), and let T > 0 be fixed. Then for all e > O, there is h(E) > O, such that for all h, 0 < h <~h(e), there is a solution cob(t) of (Eh) such that:
lifo(t)- oJ'(t)llLZ(X)
,larunt
in
10, TI.
As we will see later, the measures /Zh on Fh(12) associated with this approximation are not easy to handle, but it appears that a slight change in the approximating dynamical system improves greatly the situation with a view to our thermodynamical purpose. Let us denote by Qh the flow on Fh (S2) defined by the system (Eh). Let Ph "LI21~ Fh be the classical prolongation operator of the finite-element method (see Appendix A), and -I Yrh ~- Ph "
Let us define Lh (s -- n'h Fh (s and denote 6')~ - 7rh o Qh o Ph, the flow induced on Lh(S2). Obviously, O)/~ preserves the volume measure on Lh (S-2). From Proposition 2.1.2 we deduce the following.
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R. Robert
COROLLARY 2.1.3. Let co(t) be any weak solution of (E), with co0(x) in the space Lee(F2), and let T > 0 be fixed. Then f o r all s > O, there is h(s) > 0 such that f o r all h, 0 < h ~ h(s), there is coOh in Lh(F2) such that:
II oc, - O, OOh II
for all t in [0, T].
PROOF. By the L2-stability property of Euler equation, we only need to prove the result for coo in C ~ (Y2). Using Proposition 2.1.2, we have, for h <~ h(s): Ilco(t) - coh(t)]l ~< s, on [0, T]. Let us denote c o b ( t ) = n-hcoh(t), we have: I l c o ( t ) - coh(t)ll ~< I l c o ( t ) - rhco(t)ll + Ilrhco(t) - coh (t)II, where rh is the classical restriction operator (see Appendix A). But since
it becomes:
Now we have (see Appendix A)
chll o(')ll
C(T)h,
on [0, T],
and similarly
thus lifo(t) - OJh(t)ll ~ C ( T ) h + cs, and the result follows.
73
Let us summarize our results. We have constructed a flow (,-)]' o n Lh(~2) which approximates the Euler flow and preserves the measure dcoh - - ( ~ ) j dcoil, where coh(X) -J (x / h - . j ) (finite sum). y~j cohX 2.1.3. Baldi's large deviation theorem and thermodynamic limits. In order to define relevant statistical equilibrium states, we have to take the thermodynamic limit of the invariant Liouville measures with the conditioning given by all the constants of the motion. To perform this task we need some tools from large deviation theory. Baldi's theorem gives general conditions under which a family of probability measures on a locally convex topological vector space has the large deviation property. As we will see, it provides a powerful tool to carry out thermodynamic limits for infinitedimensional systems.
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The large deviation property. Let E be a locally convex Hausdorff topological vector space. We consider a family #h, h > 0, of Borel probability measures on E. We will say (see, for example, Varadhan [69]) that the family #h has the large deviation property with constants )~(h) and rate function L iff: (i))~(h) is > 0 and limh--->+~ ~(h) = +oo. (ii) L : E --> [0, +o0] is a lower semi-continuous functional on E (not identical to +oo). Moreover, L is inf-compact, that is: the set {vl L ( v ) <. b} is compact for all real numbers b. (iii) For every Borel subset A of E, we have:
- A ( A ) ~< l i m i n f ~ l log#h(Z), h - ~ ~.(h) and 1
lim sup
LogXh (A) ~ < - A ( A - )
h--->o~ ~
where A ( A ) --inf,,cA L(v). The functional L is also usually called the information functional, and - L the entropy functional. Let E' be the topological dual of E, endowed with the weak-star topology cr (E', E). For a Borel probability measure/z on E, we define its Laplace transform: fi(cP) -- ft; exp({cp, v ) ) d # ( v ) ,
for r 6 E'.
As it is well known,/~ is a convex, lower semicontinuous and proper functional on E'. The same is true for the functional Log/~(r BALDI'S THEOREM 2.1.4. Let ~h be a family o f Borel probability measures on E, satis.~ing the following assumptions: (1) There is a function )~(h) as in (i) such that 1
lim Log ~h (~.(h)q)) -- F(cp), h ~ ~.(h) where F is a convex, lower semicontinuous and proper functional on E' which is finite on a neighborhood o f the origin. (2) Compacity assumption: For every. R > O, there is a compact set K R C E such that 1
lim sup ~ Log #h ( K " ) <~ -- R. h--,~ ~(h) R. Let us denote by L the Young-Fenchel transform o f F, that is: L ( v ) = sup ({r r E'
v)-
F(cp)),
.for v ~ E.
12
R. Robert L is a convex, lower semicontinuous, and proper functional on E. Baldi's theorem states that under the assumptions (1) and (2) the upper bound in (iii) holds. If we suppose that L has some additional strict-convexity property, we can also derive the lower bound. We will suppose that L satisfies the following condition. (3) For every real number r, the set Ar = {v I L ( v ) <~ r} is the closure o f the subset o f the points v o f Ar where the subdifferential 0 L ( v ) is nonempty and contains an element %osuch that:
L(u') > L(u) + (tp, u ' - u),
for all u' ~ u.
Then Baldi's theorem asserts that under the hypotheses (l), (2), (3) the two bounds in (iii) hold. So, we see that the family ~h has the large deviation property with constants ~.(h) and rate function L. Indeed, one easily checks that the functional L is inf-compact on E: for every real number b , the set A/, is closed and the lower bound applied to the open set K 'b+l yields At, C K/,+l (with the notation of (2)).
Comments. (1) In practice it may be difficult to check that the hypothesis (3) is satisfied. In fact, Baldi's proof works as well with the following weaker hypothesis (3'). (3') For every v such that L(v) < + o o , for every open set O containing v and every s > 0, there is vl c O such that L ( v l ) <<. L ( v ) + s and L is strictly convex at vl, that is: ----]~pc i)L(vl) such that
L(,,') >
,,' #,,,.
(2) L is strictly convex at v if, fl)r example, i)L(v) is nonempty and
for a l l 0 < t < I , v ' 6 d o m L , v':~v. (3) In the case where only the hypotheses (l), (2) are satisfied, as we have seen, Baldi's theorem gives an upper bound. But the functional L may fail to be inf-compact in that case. Nevertheless, we can see that the set A0 -- { v E E I L (v) -- 0}
is nonempty.
Notice first that we obviously have F ( 0 ) = 0 and since F is also the Y o u n g - F e n c h e l transform of L, we get: inf L(v) = O.
vEE
Furthermore, we have g . i ( K i ) + l t j ( K ' l ) -- 1 and from (2) we know that l~h(K'l) --+ 0 (when h --+ oo).
Statistical hydrodynamics (Onsager revisited)
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Now, if A0 were empty we should have A (Kl) > 0. Then, applying Baldi's theorem we should have (/Zh (K1) ~ 0); this would yield a contradiction. Moreover, one can easily deduce that for any open set U containing A0 there is a number ot > 0 such that:
#h(U':) <~exp(-X(h)c~),
for h large enough.
We shall say that the family #h concentrates about the set A0.
Thermodynamic limits and the concentration property. When dealing with thermodynamic limits one usually encounters the following situation, which we resume here in an abstract form. Let 3h be a family of random variables with values in a Hausdorff locally convex topological vector space E. 6h generally comes from some finite-dimensional approximation of an infinite-dimensional system. If we can prove that, for h large, with a high probability, 3h remains in a neighborhood of some points v* of E, then v* is the equilibrium state of our system, and the thermodynamic limit is performed. Large deviation theory shows that such a situation is very common. We will assume in the sequel that the family 6h (or the associated probability distributions/Zh on E) has the large deviation property with constants s and rate function L. Since Prob(gh ~ E) -- 1 for all h, we have inf,,~- L(v) -- 0 and the set A0 is a nonempty compact subset of E. And, as in the above comment 3, for every open set U containing A0, there is some c~ > 0 such that: Prob(ah 6 U") ~ e x p ( - 1 ( h ) o t ) ,
for large h.
That is, the family 3h concentrates about the set A0 which is the equilibrium set of the system. In our "microcanonical" approach, we will study now the situation where &l satisfies some given constraints (it would be more correct to say that we introduce some conditioning on the random variables 6h). These constraints will be given, for example, by the constants of the motion of an infinite-dimensional dynamical system. We introduce the constraints in the general form 3h 6 s where s is some subset of E. Of course, since 3h comes from a finite-dimensional approximation, the ideal constraints 6h 6 s will not be exactly satisfied, but only up to some approximation given by an open neighborhood of 0 in E, W. Let us denote s = s + W. We shall then consider 6h 6 s Let us now give a definition. DEFINITION. Let g, s be subsets of E. We will say that 6h concentrates about g* conditionally to g iff: I Log Prob(3h E s > -oo (i) 'v'W', lim infh--,oo z--~
(ii) 'v'W*, 3or > 0, 3W, VW', Prob(3h 6 s
\s
Prob(6h E Cw,)
)
~< e x p ( - 1 ( h ) o t ) ,
for h large enough.
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R. Robert
Here W*, W, W I denote open neighborhoods of 0 in E. REMARKS.
(1) Heuristically, this definition means that if we know that 6h takes its values in a neighborhood of E, then it will be in a neighborhood of E* with a high probability. (2) As previously noticed, we have to widen the sets ,5', E* into open neighborhoods. In fact, Prob(3h 6,5') is not defined for an arbitrary subset C; and even if E is a Borel subset, it can be zero. (3) The condition (i) ensures that, when h ---> cx~, Prob(ah ~ Cw,) cannot be too small. Now we derive the following concentration result which will be useful to carry out thermodynamic limits. CONCENTRATION THEOREM 2. 1.5. We suppose that 6h has the large deviation property
with constants X(h) and rate function L. Let C be a nonempty closed subset of E and E* the subset of E where L achieves its minimum value on E. Then 6h concentrates about C* conditionally to C. PROOF. See [41 ].
1-3
REMARK. The set ~'*, in which 3h approximately remains with a high probability, is the equilibrium set of the system. If ~'* does not reduce to a point (the equilibrium state), we are in a phase transition situation. We shall see in the following how we can use this concentration result to derive a maximum-entropy principle for Young measures. 2.1.4. A maximum-entropy principle for Young measures. It currently happens when dealing with a limit process for a sequence of bounded measurable/'unctions that the sequence does not converge and shows an oscillating limit behavior, whereas some estimates and conservation laws hold. In such a case, the concept of Young measure has been found relevant to describe the behavior of the sequence (examples can be found in hyperbolic systems of conservation laws, homogenization, h y d r o d y n a m i c s . . . ). Young measures can be viewed as giving a macroscopic description of the system, whereas the bounded measurable functions are all the microscopic states. We use the results of Section 2 to derive a maximum entropy principle for Young measures, that is: the macrostate (Young measure) which realizes the maximum of an entropy functional has a natural concentration property (a large majority of the microstates satisfying a given set of constraints are in a neighborhood of that macrostate). It turns out that this entropy functional is the classical Kullback entropy (see Sanov's theorem).
Young measures. Throughout this section, X, Y will denote two locally compact separable and metrizable topological spaces. Let us suppose that a positive Borel measure dx is given on X. Let us recall that Young measures [71] are a natural way to generalize the notion of measurable mapping from X to Y: at any point x 6 X, we no longer have a well-determined value, but only some probability distribution on Y.
Statistical hydrodynamics (Onsager revisited)
15
In other words, a Young measure v is a measurable mapping x ~ Vx from X to the set Ml (Y) of the Borel probability measures on Y endowed with the narrow topology. Clearly, v defines a positive Borel measure on X x Y (that we will also denote by v) by:
(v, f ) -
fx(VX, f (x, . ))dx,
for every real function f ( x , y), continuous and compactly supported on X x Y ( f Cc(X x Y)). Moreover, for f ( x ) ~ Co(X), we have
(v, f ) - f x
f (x) dx,
that is, the projection of v on X is dx. It is well known [41 ] that this property gives an equivalent definition of Young measures. That is, for any positive Borel measure v on X x Y whose projection on X is dx, there is a measurable mapping x ~ Vx such that the above formula holds. The mapping x --+ Vx is unique up to the dx-almost, everywhere equality. To any measurable mapping f " X -+ Y, we associate the Young measure 6 f ' x --+ ~.f(x), Dirac mass at f (x). We shall make two additional assumptions" (,) The measure dx is diffuse and of finite total mass. (**) There is a distance function d(x, x') giving the topology of X, such that" for all s > 0, there is a finite partition of X into measurable subsets X = {X i [i -1. . . . . n(3~)} with I x i l - IXJl for all i, j (we shall say that 3~ is an equipartition of X), and satisfying d(,t~) ~< s, where d(3~) - sup/SUpx.x,~Xi d(x, x') is the diameter of Y. Notice that (,) and (**) imply that IA [ approaches zero when the diameter of a measurable set A approaches zero. Hypotheses (,), (**) are satisfied, for example, if X is an open convex and bounded subset of R" with dx -- Lebesgue's measure, and also if we consider any image of X by a dx-preserving homeomorphism. We shall denote by M the convex set of Young measures on X x Y, and we recall some useful properties. 9 M is closed in the space Mh(X x Y) of all bounded Randon measures on X x Y (with the narrow topology), the narrow topology is equal on M to the vague topology (weak topology associated with the continuous compactly supported functions) and it is metrizable. Furthermore, if Y is compact then M is compact. In the sequel M will be endowed with the narrow topology. 9 {~f[ f ' X ~ Y measurable} is a dense subset of M. This property is classical in the case Y is compact (see reference in [41 ]). The general case follows by approximation. Approximate first (for the vague topology) a given Young measure v by vx (as in the proof of Theorem 2.1.6 below) which is constant, equal to v i, on each set X i of an equipartition X, and then approximate each v i by a probability measure with compact support.
!6
R. Robert
A large deviation property. Suppose now that we are given a basic Borel probability measure n'0 on Y. Then with any equipartition 5E of X we can associate a Borel probability measure #:~ on M in the following way. We take Yl . . . . . )~n, n(3~) Y-valued independent random variables with the same distribution n'0. We consider the random function f.~ -- Z
y i 1xi, i
where 1xg is the characteristic function of the set X i . We denote by 3x the Young measure associated with f.~, and by # x the probability distribution on M of the random variable 6x. Now, we can state the main result of this section. THEOREM 2.1.6. When d(SE) ~ O, the family #:~ has the large deviation property with constants n ( Y ) / I X [ and rate function Ijr (v), where 7r - dx | 7ro and Ijr (v) is the classical Kullback information functional (see Varadhan [69]), defined on M by: Irr (v) --
{
f x ~ v Log ~dt, dr,
if v is absolutely continuous with respect to Jr, otherwise.
PROOF. The proof is an application of Baldi's theorem, see [41 ] for details.
[-I
In this Young measure framework, Theorem 2.1.5 yields the ff)llowing. COROLLARY 2.1.7. Let C be a nonempty closed subset of M, r the subset o[C where the.f'unctional l~r achieves its minimum value on C. Then 3x con~'entrates about E* conditionally to C. REMARKS. 9 Note that since the functional/Tr is inf-compact and ,5' is closed, ,5'* is nonempty. 9 Theorem 2.1.6 appears as a generalization of the well-known Sanov's theorem. Indeed, apply the contraction principle to the mapping v ---> f vx dx. 2.1.5. Thermodynamic limit of the invariant measures in the space of Young measures.
Long time dynamics and Young measures. As we have seen, Euler system describes the advection of a scalar function (the vorticity) by an incompressible velocity field, thus the vorticity oJ remains bounded in L~(,q2). The functionals
Co ( o 2 ) - ff2 O(~o)(x)dx, are constants of the motion (for any continuous function 0). That is to say, the distribution measure of co, trio, defined by (trio, 0) = Co (co), is conserved by the flow. Let us consider an initial datum coo. It is well known that, in general, as time evolves, F~o~0 becomes a very intricate oscillating function. Let us denote r = [[~o0i[L~(~). Since the
Statistical hydrodynamics (Onsager revisited)
17
measure n'o~ is conserved, Ftcoo will remain, for all time, in the ball Lr~ -- {co: I1~o11~~ r}. Extracting a subsequence (if necessary), we may suppose that, as time tends to infinity, Ftco0 converges weakly (for the weak-star topology o'(L ~ , L1)) towards some function co*"
Ftco0
w > 09, 9
We can easily see that Co(l-)coo) does not converge towards Co(w*) if 0 is nonlinear, whereas some other invariants can converge, as it is the case for the energy. So, much information (given by the constants of the motion) is lost in this limit process. Thus, the weak space L ~ (f2) is not the good one to describe the long-time limits of our system. Fortunately, the relevant space to do this is well known, it is the space Mr of Young measures on f2 x I - r , r], that we have just defined. We can identify the long-time limits of the system as Young measures. Indeed, Mr is a suitable compactification of Lr~ since the narrow convergence (when t approaches infinity) of 6~oo towards some Young measure v preserves the information given by the constants of the motion, that is, for all functions 0(z):
fs 0(r,~o()(x))dx -+ fa <~x,e>dx, but the left-hand side is constant and equal to (Tr,,,0" 0), so that:
fs2
vx dx - 7r,,,0 .
(,)
The same arguments apply to the other invariants. For example, since Ftco0 converges weakly towards fi(x) -- f z dvx(z.), we have, for the energy, ,T(Ftco0) -+ ,~(~), which is the energy of the Young measure v, and thus: = (~) -- ,.,=(coo) We shall denote by (**) the set of constraints (associated to the constants of the motion) other than (,), that fi has to satisfy: = {energy constraint, angular momentum constraint (eventually)].
(**)
Thus we see that the constants of the motion bring the constraints ( , ) , (**) on the possible long-time limits. Since we do not know anything (in the general case) on the long time behavior of the solutions of Euler equation, we will consider Young measures merely as a convenient framework in which we can perform the thermodynamic limit of a family of invariant measures. In the following, we will call the Young measures satisfying the constraints (,), (**) the mixed macrostates, in contrast to the small scale oscillating vorticity functions called microstates.
18
R. Robert
The random Young measure ~fh" Let s be a bounded open subset of R d, the space Fh (s is composed of the functions of the form y~j f J / 3 ( ~ - j ) which are compactly supported in s (see Appendix A). The space Lh (Y2) = Zrh (Fh (s is composed of functions of the
form Z j f/{ X(~ - J) which vanish in a neighborhood of the boundary 0s (whose width 9
X
approaches zero with h). Let us write a function of
(~j~jhdf/, and # h -
89
Lh(g2)" f h - Y~j~Jh f / , x ( - f i - J). We denote d f h f fZdx)dfh, the probability measure on L h ( ~ ) , 9
X
where the scaling factor 1/h a is introduced in order to give a finite value to the mean
fLh(S2)(f f2 d x ) d # h ( f h ) , in the limit h ~ 0.We will write/Zh -- (~)j~Jh dzr*(fhj ) ' where dTr,(y) -- _1__1 vq- e-y2 dy. We will consider now fh as a random function with probability distribution #h- Thus ~i/i, is a random Young measure on s x I~. It follows from Theorem 2.1.6 that the family (depending on h) of the random Young measures ~i/), has the large deviation property with constants 1/ h a and rate function I~ (v), where we denote 7r -- dx | Jr, A straightforward consequence of this large deviation property is that the random Young measures ~i/), which in addition satisfy the constraints (,), (**) are exponentially concentrated about the set g'* of the solutions of the variational problem l~(v*)--inf{l~(~):
~ c ~ ' l,
where g" is the closed subset of the Young measures on .(2 x R satisfying the constraints (,),
(**). Note that this variational problem has at least one solution since ,5' is nonempty and closed and l~r(v) is a lower semicontinuous and inf-compact functional on M. Now, let us denote 7r0 = ~ 1 zrr 0 and 7r f - dx | 7r0. For all v satisfying (,), one can easily get the relationship: l~(v) = l:r,(v) + If2l/zr, (n'0). Thus if G, (Tr0) < oo, minimizing /zr or Izr' on g' gives the same equilibrium set g'*. In fact, the use of the functional /~r, is more natural since it is associated to the invariant distribution 7r,~0. To justify the use of 17r' in the degenerate case/~r, (Tr0) = e~, one can, for instance, modify the definition of the measures Uh, and consider Uh -- (~.i~Sh dTrh (.[)[ ), where dTrh(y) -- 7l exp(--Qh(v))dv, and the polynomial function Qh(Y) is such that zrh converges towards zr0 in the narrow topology when h ~ 0. Of course, we have
/th--~exp
' ( -'- ~f-
Q /, (.fh) dx)
dr),.
It is not difficult to see that the proof of Theorem 2.1.6 [41 ] works for these measures, and it follows that 6./), has the large deviation property with constants l / h a and rate function l~r' (v). Notice that - I j r , ( v ) is the entropy, that is, the functional which measures the disorder created in the fluid by the turbulent mixing.
Statistical hydrodynamics (Onsager revisited)
19
REMARKS.
9 For Euler equation we have d = 2. We shall consider also the case d = 6 for V l a s o v Poisson system. 9 In order to get probability measures, we replace d fh by phdfh with Ph = lZ e x p ( - - ~ f 2 dx) despite the fact that this functional is (eventually) not conserved by the flow. Indeed, we consider as an authorized trick to multiply the measures by any functional which is conserved by the flow of the infinite-dimensional dynamical system.
f
2.1.6. Computation of the equilibrium states, the equation of Gibbs states. Once we have identified the relevant entropy functional, the determination of the equilibrium states come down to the solution of a variational problem: i.e., find the minimum value of Izr'(v) under the constraints given by the constants of the motion of the system. After that it remains to discuss at a physical level the relevance of these states. From now on (for obvious typographical reasons) we shall denote co instead of coo, rr instead of 7r', and K~r(v) = - G ' ( v ) the Kullback entropy. We have now to solve the variational problem: Find the macrostates v* satisfying
where co is any initial vorticity in L~(S-2). We have seen in a previous remark that such v* always exists. We begin with the simpler case where co takes only a finite number of distinct values at . . . . . a , (co takes the value ai on the set S-2i). Then, we have 7r~o = IS2113,~ + . . . + IS2,, 13,,,. It is clear that any mixed macrostate is of the form vx = el (x)6, t + - - . + e,, (x)6,,,, with the constraints:
Fi (el . . . . . e,, ) -- fs2 ei (x) dx -- IS'2il,
i=
l,...,n.
The most mixed state is such that ei(x) --IS2il/IS21, for every x" that is, zrx : T-~-Tzr~,,. The probability distribution Vx is obtained by multiplying rrx by a function equal to ei(x)]F2l/lI-2il for a = ai. This function is equal to dv,:/drrx. Therefore, the Kullback entropy writes as: i
Kjr(v) -- -
~
ei(x)
i
Z
Log
i--~/]ei(x)
t
ei (x) Log ei (x) dx - Z i
dx
IS21).
1s I Log ~ i
i
R. Robert
20
Since the second term is a constant, this entropy is indeed equivalent to the classical Boltzmann mixing entropy"
S(e)-- - f ~ Z
ei (x) Log ei (x) dx.
As e = (e~ . . . . . en) must satisfy the supplementary constraint Z i L 1 ei(x) -- 1, only n -- 1 independent constraints Fl . . . . . Fn- l remain. Let e *I , . . . . e n* be a solution of our variational problem. Then, by the rule of L a g r a n g e multipliers, there are constants c~ -- (c~l . . . . . Otn-l) and fl such that the first variations of the functionals satisfy: n-I
i--1 for all variations 6el such that Y'~i ~ei -- O. Straightforward computations give:
6 s - - - fs2 Z ( l i
+ Loge*)3eidx,
6E -- -- fs2 ~* (~i
ai6ei) dx'
where ~* is the stream function associated with the vorticity ~--~iaiei"*(x),
Fi -- is-2 ~ei dx. Then, we easily get:
e*(x) --
e-Oti -flai Vz*(x ) (l)
z(~*(x))
where the partition function Z is given by
Z(~) = ~ e -~;-~'';v' i=1
and we take C~n -- 0, by convention. Thus ~ * satisfies the equation of Gibbs states:
(G.S.E.)
- - A ~ * = ~--~iaiei*(x)= ~p* - 0 on 0S2.
d Log Z(~p*), fll Or
}
So, we see that if e i* is a solution of the variational problem, there are constants c~, fl such that e* is given by (1) and 7t* by (G.S.E.).
Statistical hydrodynamics (Onsager revisited)
21
Conversely, for any given set of parameters a,/4 we can consider a solution gr c~'/~ of the nonlinear equation (G.S.E.) (as the right-hand side of the equation is of the form .f(O*), with f continuous and bounded, we know, using Schauder's fixed point theorem, that a solution, in general not unique, always exists) and the associated Gibbs state e ~ given by (1). Of course, e ~ is a critical point of the functional J ( e ) = ,n- 1 n S(e) - - ,fl_ ,v ( e ) -- rz-,i=l oti Fi (e) , on the linear manifold E i = l e i (x ) -- I. Furthermore, we prove the following result: P R O P O S I T I O N 2.1.8. /ffl > --Xi/(}--~ia2), where )vl > 0 is the first eigenvalue ofithe ope r a t o r - - A (associated to the Dirichlet b o u n d a r y value condition), then e ~'fi is the unique m a x i m u m o f the.functional J (e) on the set defined by ei (x) >~ O, i = 1. . . . . n, ~ i ei (x) = 1.
PROOF. We shall prove that the functional J (e) is strictly concave on the set defined by 0 < e l ( x ) < 1, i = 1. . . . . n. For that, let us compute the second variation 32j for any
variation 3ei. Straightforward computations give: 1 fs2~-~(fei)2 f'~ S --
2 i=!
f 2 , T , - l fs 2 fig, 6 w d x 2
ei
where we denote f w = E i aifei and fV,~ is the stream function associated to fw. As f2 Fi = 0, it becomes: 32j=
lf2~(fei)2 2 ei i=1
dx - tiff-2] ; 5vl 3wdx.
Let us consider first the case fl ~ (). Then we have, by Green's fl)rmula:
fs2
3 5 3wdx -- ] ~ (V3~)2 dx >~ O,
from where:
l/2~(3ei)2 32 Y <~ - - 2 ei
dx < O,
i=1
thus J is strictly concave. We consider now the case/4 < 0. Using the well-known inequality
~
6 w d x <~ -s
(6w)- dx
o
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Statistk'al h vdrodvmtm i~'s (Onsager revisited)
25
2.2. Extension of the theory to other ,2vstems, the Vlasov-Poisson equation 2.2.1. A c/ass ~?fdynamical systems. A large class of evolution equations, coming from the modeling of various physical phenomena displaying complex turbulent behavior, can be described as the convection of a scalar density by an incompressible velocity field. More precisely, they are of the form:
{
ql + div(qu) - 0, u - - L ( q ), div(u)--0,
where q(t, x) is some scalar density function defined on R • S2 (S2 is a bounded connected smooth domain of R~/), u(t, x) is an incompressible velocity field taking its values in R ~/, which can be recovered from q by solving a RD.E. system. Thus L denotes a (not necessarily linear) integro-differential operator. Let us give some well-known examples of such systems. (1) The simplest example of (1) is the linear transport equatioll, where u(x) is a given incompressible velocity field on ,<2. (2) The quasi-geostn>phic ntodel of geophysical fluid dynamics [40]:
{
q~ + div(qu) - (), u -- curl q,', -At/,, + ~.q~ - q + .l(x),
q,' = () on i)
where q is the so-called potential vorticity, c is a nonnegative constant, and ./ a given function in L"~ (S-2). Obviously, Euler equation is obtained by taking c and ./ equal to (). (3) Collisionless kinetic equations such as the Vlaso~,-Poisson eqttatiott of stellar dynamics can also be written in the form (1). Indeed, let f(t, x, v) be the scalar density function giving the distribution of the stars in a galaxy (in the phase space (x = position, v = velocity)). If collisions (close encounters) are neglected, the density is purely advected by the flow in phase space, so that it satisfies the following Vlasov-Poisson equation:
(V)
af + divx(v 1) + d i v v ( E f ) -- 0,
where E(t x) is the gravitational field, E ,
-Vxq> @(t x) - - G .[' P(t'x' I"dx' is the grav,
,
']X_X
~9
itational potential, and p(t, x) = f f ( t , x, v)dv is the spatial density of stars. Equation (V) conserves the following functionals: The total energy (kinetic + potential): -
,~(.f)--
I/Iv2 ~
(.f(x,v))dxdv+~ If
p(Pdx;
R. Robert
26 -
The integrals
fr -
The linear momentum
ff -
o ( f ) dx dv, for any continuous function 0 such that 0(0) -- 0;
v f dx dv"
The angular momentum
ff
x A v f dx dv.
Let us d e n o t e U may be written:
(~,';)" U is a vector field on R (', which satisfies d i v e , ( U ) -
.1} + div(,(./U)
O, a n d ( V )
= (),
so that Equation ( V ) in of the form (I). The lirst step in our program is to deline a tlow associated to (!) on the phase space L ' (,(2). Untk)rtunately, to our knowledge, there in no general existence-uniqueness result for the Cauchy pn~blem lk~r systems like (l). Examples I and 2 are well known, bt!! c()nccrning Vlasov-R~isson equation, things are more intricate. One can lind existence results for regular initial data in [6 ], while the existence of weak solutions in investigated in [3,2(),28,37]. A nice existence theorem of weak solutions corresponding !c~ a~ initial density ./i~ in I,~(I~ ~') ~ 1,*(1I~'), satisfying ./j'~Iv2 ./ijdxdv < cxs, is given by Horst and Hunze[28]. An usual, uniqueness in more tricky, and we need some additional assumption on the initial density. Let us mention two recent results. - with a Lipschitz regularity condition on ./i) (Lions and Perthame [37]), - with ./i) in L ~ and compactly supported (Robert [5()]). Finally, for our concena, if ./i) in given in I, ~ with compact support, the situation in completely analogous to that given by Yudovich's theorem for Euler equation: We have a unique weak solution .f(l, x, v) in ('([(), ec[: L/'(R(')) for all / 7, I ~< p < cyo, which delines a ttow FI on L'V (R(~). Using the moment estimates of [37], we easily prove that this weak solution remains compactly supported tk)r all time. Here we have also a stability property" I1" ./i~ in a bounded sequence in L < ( I U ' ) , all the functions having their support included in a same compact net which converges in the strong L2 topology towards ./i), then F~./i~ uniformly converges L2-strongly towards Ft./i) on any bounded time interwd. 2.2.2. Fillite-dimensionalapln'oximatiotls o./'Vlasov-Poisson equation. We will consider weak solutions of (V), on a fixed time interval 10, T], corresponding to ./i) in L~(R(~),
r-
r-
I
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R. Robert
34
The M.E.RP. then gives the following variational problem (V.P. 1):
(J! . . . . . J,,) - max } (JI . . . . . J,,), under the constraints: (C l) ~ i Ji (x) = O, for all x,
(C2) ~" (Jj . . . . . J,,) = 0,
j2 (C3) _ ~ - - ' ( x ) ~ < C ( x )
for a l l x
It can be shown [53] that this problem always has a solution Jj . . . . . J,, satisfying (C3) with equality; moreover, there exist a parameter/4 and a measurable function A(x) such that:
(A)
Ji - - A ( x ) [ V p i
- fi((7)- ai)piV~/I].
This gives a general fl~rm fl~r the currents JI J,,. The variable (~> O) diffusion coefficient A(x) is not known; but once it is given, the parameter fi, which is the Lagrange .
.
.
.
.
multiplier of ,E,, is determined by the conservation of the energy"
/
V (,, 9,I,,, dx - (),
which gives
(B)
/4---
V~/J. VrT)A(x) dx
/L
(Vt/i)2((,)- 2 - ( 7 ) 2 ) A ( x ) d x ,
where o) 2 - ~--~-i ct'[ Pi. Next we need to determine the diffusion coefficient A (x). For this, let us consider the particular case fl = (}. in this case, the energy constraint is not active and Ji is merely an ordinary diffusion current. Thus, we can compute A (x) by using the analogy with the convection-diffusion of a passive scalar. Let us introduce the notations: w=&+&,
u=fi+fi,
& is the fine-scale fluctuation of the vorticity and fi the corresponding fluctuation of velocity. Let us suppose that some scalar density p(t, x) is convected by the microscopic flow u. Then the mean value fi(t, x) of p(t, y) on some ball B(x, r) is convected by the mean field fi and undergoes a diffusion process created by the fluctuation ft. /~(t, x) will satisfy an equation of convection-diffusion type: tst + v . ( ~ s f i + J ) = 0 ,
II
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Ci.
-,.
36
R. Robert
The link between ( R E n ) and the equilibrium theory is given by the following convergence result. PROPOSITION 2.3.1. Let us suppose that the solution p i ( t , x ) , i - - 1. . . . . n, of ( R E n ) converges (in a strong e n o u g h sense), when t tends to infinity, towards a stationary state Pi*(x). Let us a s s u m e also that there is some open c o n n e c t e d s u b d o m a i n A o f F 2 satisfying A C F2 and: (*) Pi* (x) > 0 and A (p*)(x) > O, f o r all x in A. (**) U* 9n -- 0 and J* 9 n - 0 on OA. (n is the unit vector n o r m a l to the b o u n d a r y OA, u* is the velocity field associated to Fo, -- ~ aip*, a n d J~ is the current a s s o c i a t e d to p* by ( A ) . ) Then Pi* (x) is a Gibbs s t a t e on A" that is, there are p a r a m e t e r s ot I , . . . , or, (otn -- O) a n d fl* such that:
Pi
*
(X)
--
exp(-cF - fi*ai~*(x)) Z( ~* (x))
moreover, fl* - - l i m ; _ ~
,
'
for all x in A"
fl(t), where fl(t) is given at each time by (B).
PR()OF. When t --+ cx~, we have: p i ( t , x ) -+ p*(x), &(t,x) --~ &,(x), ~//(t, x) --* r and fl(t) --+ fi* (given by (B) where we replace p i ( t , x ) by p*(x)). We have also Ji(t) -- A ( p ( t ) ) [ V p i ( t ) - fl(t)(rb(t) - d i )pi(t)V~/J] ~ Ji*, and since p i ( t , x) satisfy (REtt), the t'unctions /if (x) will satisfy the set of stationary equations" V . (/,*u* + J * ) - ( ) ,
i-I
. . . . . ,,.
Now let us calculate fA E
Ln I , * V . (/,*u* + Ji*)dx i
-L
E
Ln p* V p * . u* dx + f,a E i
Ln p* V. J* dx. i
Integrating by parts, we see that the first term is zero (due to V 9u* -- ()) while the second gives"
v P7
9Ji* d x -
-
-
fA~ fi*
9 P---~i I[Vz,
fA
* - /~*( , - b , --
ai
)V~*-J*
a i ) l i,* VO*]
9Ji9 dx
dx,
I
but the last term vanishes since y~ J , - 0 and J~a V~p*. J,.* dx - 0 (this last equality comes from .['A ~ * V . (/flu* + J * ) d x - O, by integration by parts).
Statistical hydrodynamics (Onsager revisited)
37
We finally get
fA Z
i
--Z-Z[ 1 Vp* -- ~*(w* --ai)Pi, V ~ * ] A ( p * ) d x - - O, Pi
from where V Ln Pi* -/4*(&,
-
Cli)V~* - - 0
on A,
for i -- 1. . . . . n.
Subtracting equation n from equation i, we deduce that L n ~P; + # ,
(ai --an)~ 9
has a
constant value -or; on A. Now, using the relationship ~ p*(x) -- 1, we deduce that p* satisfy the Gibbs state relationships on A. D The relaxation equations (REn) have been used for large eddy simulations [53]" they where found very efficient.
3. Out-of-equilibrium problems" Weak solutions, shocks, and energy dissipation 3.1. Statisti~'al solutions of ID invis~'id Burgers equation 3.1.1. Definitioll qf statisti~'al s<>httiolls
dt
=a(u),
u(O)--u(),
Let us begin with the
(I)
where a" R" -+ IR" is a smooth mapping, and we assume that for any given initial state (I) has a global smooth solution defined for all time"
u(t)
-
-
%u().
The family Ot is a group of smooth diffeomorphisms of Rn. Then to any given initial Borel probability measure/~() on R" we can associate the family
I~I -- ~t (~()).
(II)
Thus, in this case we clearly know what we mean by statistical solution" a family of Borel probability measures satisfying (II). Let us now introduce the characteristic function
38
R. Robert
Making appropriate integrability assumptions, we calculate the derivative 3,fi, (v) -- i
f
a ( u ) . r e ' " " d/tl(u),
for all v in IR".
(III)
We can easily check here that, under some integrability assumptions, (III) is equivalent
to (II). Let us now consider the more general case where (I) is an ew)lution equation, u taking its values in some functional space. Let us assume that the Cauchy problem for (I) can be solved by means of a continuous semigroup St acting on the space E (with dual E'). The same calculation as above for/it = St (P0) formally yields:
f<.,,,,.
,,, ,,
dp, (u),
for all v in E'.
(III')
We will consider the fl)rmula (III') to give a definition of statistical solutions. More precisely, since we will always consider for E a dense subspace of the space of distributions 79'(R"), we will have CI~ C E'. DI':I:INITI()N. We shall say that t*~ is a statistical solution of(1) if fl~r all ~5 in (7r fit(z~) is a C I function o f l, the right-hand side o f ( l i t ' ) is properly detined and the equality hc)lds. REMARK. At this level of generality, the question of the uniqueness ot" the solutions of (i11') is open. Indeed, even if we suppose the existence of a semigroup $I, we certainly need strong assumptions to prove that a solution of (i11') satisties t*t = & (I*{~).
3.1.2. k)'om ho,u~etzeotts to illtrilzsi~' statisti~'ai s o h t t i o l t s q / B l t r g e r s eqttatiolz.
The one-
dimensional Burgers equation
(B)
iJ t II -~- iJ.v
('
-~ ll
-- ()
!
is of the form (I) with ti(ll) -- --iJ.v(-5l!2). Equation (B) has been extensively studied [58]. We know that starting with a smooth initial data tt(). a unique smooth solution exists for a short time but discontinuities may occur at a linite time. We can also prove that discontinuous weak solutions exist for all time but then uniqueness is lost. To ensure the uniqueness of such weak solutions one has to introduce a supplementary condition (Lax entropy condition which amounts to consider only discontinuities with negative jumps). Finally, Kruzkov's theorem gives a contraction semigroup in the space L I lk~r the weak solution. After integration by parts (III') gives for the general statistical solutions
~}t/~,(V)
-- i
-~u-
ei
") d p , ( u ) ,
for all v in C ~
Statistical tlydrodynanlics (Onsager revisited)
39
where #~ is a family of Borel probability measures on 7)'(IR). For any real number h we denote rh :7)' --+ 79' the translation operator r h ( f ) ( x ) = f ( x -- h). We shall say that/z~ is an h o m o g e n e o u s statistical solution if, in addition, we have for all t: Z'h(ftr =/zr
for all h.
A rather straightforward but important remark is that if #t is a h o m o g e n e o u s statistical solution of (B), it is also a statistical solution of
('
)
ibu + 'a,. -~(u - c) 2 - 0 ,
for all c.
This remark brings us to the following useful proposition: PROPOSITION 3. I. 1. Let t*~ be a homogeneous statistical solution o [ ( B ) sati.sfving (i) j' [Ittl[21.2(,I ) dl~t(u) < oo,.fi)r all t and all compact intervals J" ,
(ii) .l'(u u,
t)2
9
d/zt (u) <~ c..l" w- dx, for all w in C ~ . Then, for any.l"un,'tio,~ p in C~7~ with integral 1, and p , , ( x ) -
a,fi,(,,) ilimf(l(,, (,,p,,))2) --
,,~
F_
--
, v'
,
e
i<,, ' ,,> d t , e l
,Lp( ~,), we have:
(ll) ,
.for all v i~l Cir.
PR()()F. We have -~
--
,
-~lt - , v '
,
e i
,,
)
(.)
e i(,, ",,) d t ~ 1 ( l l )
-
/
(it , p , , ) ( l t
,
, v )e '(''''')
dlt,(lt)
,
and we now prove that
f ( , , , p,,){u, v'}e i<'''''} dt,, (1,)
~. (). 11---+4
For this, let us consider the function ./(11) = f ( r h u , p,,)e i (r/,,,.,,) dt*1 (u). Since l,tr is honlogeneous, ./ is independent of/1, taking the derivative at/1 -- 0 yields"
f(,,, p,;)e'<"."> I,,)+if <,,.,,,,>(,,. ,,')e'<"."> (,,/-0. Thus, we only need to prove that > 0, n--* + r
-2-
s
oo
""
~
,r
~
=8
~.
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Statistical h win)dvnamir
( Onsager revisited)
41
where we use the notation (u | u : Vv) = (uj uk, OJvk ). The definition of intrinsic statistical solutions then straightforwardly follows: 0 , f i 1 ( v ) - i ,,~or
(u, ,o,, )) | (u - (u, ,o,,))'VP(v)}e i('''v} d/z1(u),
f((u-
for all v in (C0~C~)3. We will now exhibit a class of intrinsic statistical solutions of Burgers equation. To do that, we need some material from the theory of Levy processes. 3.1.3. A rough presentation of LdD, processes. For a clear and comprehensive study of Levy processes we refer to [10,27]. We only want to give here an alternate presentation from the general point of view of probability measures on functional spaces. We define directly a homogeneous Levy process with finite variance on the line as a Borel probability measure t* on the quotient space 7)'(R)/constants, with characteristic functional (defined on C0{ ~ ) of the form
a,,,, where (i) (ii) (iii)
(f r
,) a,),
I ~,(.v) -- ./'x 1,(.v) ds, and the function ~//9R --, C satisfies t/i in C 2 and ~/i(())= (), (/(-u,) -- (J(u,), ~/,, is conditionally of positive type, or equivalently -~/,," is of positive type.
The function ~/, is the Levy exponent of the process. Elementary calculations using Bochner's theorem yield the estimate
]r
- ~/,'(0)~1, I ~< ~ ~,2.
Let us consider the functional
fr
l'or u~ in Ci~"
C is obviously continuous on tile space Cir. It is classical to check (usin,,e the properties (i), (ii) (iii)) that it is of positive type and thus by Minies' theorem it defines a Radon probability measure v on the space "P' 1271. The linear operator / is continuous from C{I~ into Ci~, and its transpose I1"'D' ---+ D'/constants is the primitive operator. Thus exp(.]" ~//(1 v(x) dx)) is the characteristic functional of a Radon probability measure 1' on the space 7)'/constants, image of v by ~1, it is the Levy process with exponent ~/J. Using general arguments, we can get further regularity properties of the Levy process. For any open bounded interval J = ]a, b[, the image of v by the restriction operator to J is actually a cylindrical probability on the space L2 (,I) since C (w) is continuous for the L 2
v
+
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46
R. R o b e r t
Let 99 be any infinitely differentiable function with compact ,support on IR3, even, nonnegatire with integral 1, and ~p~:(~ ) - ~q9( 7 )" Put D~:(u)(x) -- 88J' V~0~(~) 93u(3u) 2 d,,e, where 3u -- u(x + ,e) _ u(x). Then, as e approaches O, the functions D~:(u) (which are in L I (]0, T[ • converge, in the sense qf'distributions on ]0, T[ • 7-, towards a distribution D(u), not depending on qg, and the following local equation q f energy is satisfied
)+div
) - v A ~ l u2 -+- V(VU) 2 -+- D ( u ) -
+p
0.
PROOF. Using Sobolev inclusion of H t in L 6, one easily sees u is in L3(0, T" L 3) and therefore uiuk is in L3/2(0, T; L3/2)" the same for p since, taking the divergence of (1), one gets -- A I ,
--
ik
Oi (l! i It k
),
and if p is the only solution with mean zero, the linear operator t t i t t k continuous on L q for I < q < oo, and so p c L3/2(0, T; L3/2). Now let us mollify Equation (I)" denoting u *: - ~p~,u, f f - ~p~:,l,, ( t l i U ) one has iJ t U ~ -~- iJ i ( II i U ) ~' - - I; A U ~ -JI- V
~
p is strongly
~: - - 99 ~ : * ( u i u )
....
p ~ -- ():
this equation, multiplied scalarly by u plus Equation (I) multiplied by u ~, gives
a, (u. u")+ div((u, u~)u + f l u + l,U ~) + E~-
I,A(u. u ~) + 2 v V u . Vu ~ - (),
where
E~,(t ' x) - i~i(ttitt.i )~ lt.i - grill.jill "~i" Since u e L3((), T: L3), u . u * converges to u 2 and ( u . u~)u + p*:u + p u * converges to (u 2 + 2 p ) u in the sense of distributions on 10, T[ • Moreover, Vu" tends to Vu strongly in L2(](), T[ • thus E , ( t , x ) converges in the sense of distributions towards - a t ( u 2) - d i v ( u ( u 2 + 2 1 , ) ) + v A u 2 - 2 v ( V u ) 2. Another calculation gives f
V~p~ (~) 9 6u(6u) 2 d6
: --i)i(uittjttj But Oi (ujuj)~:
+ 2i~i(ttiuj
Ui - - i)i ( u i ( U j U j ) ~ " ) ,
uj
+ (~i(ujuj )~:tli
2ttitt.ji~iu.!
due to the incompressibility of u.
~rl
=.
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0"
'.-~
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c
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R. Rohert
4g
the lack of smoothness could not lead to local energy creation. In other words, one should have D(u) >~ 0 on ]0, T[ x 7-. It is quite remarkable that this condition is satisfied by every weak solution of NavierStokes obtained as a limit of (a subsequence of) solutions u,: of the regularized equation introduced by Leray [34,35 ]:
div(u~: ) -- 0,
u~: (()) -- 9o~: 9 uo.
For uo given in L2 and s > 0, this equation has a unique C ~ solution u,. The sequence (u,) is bounded in L2(0, T: H I ) A L ~ ( 0 , T: L 2) and a subsequence converges to u, a weak solution of Navier-Stokes, weakly in L2(O, T: H I) and strongly in L3(0, T: L3). But for the regularized equation, one has the local energy balance: (1
,)
+ div
(
,.
1 ,
)
+
,u2+,,(Vu,)2
(),
-
hence ~,(Vu,, )2 converges in the sense of distributions towards
i),
2
-div
((,oe)) u
2
+/'
+l,A2
,~
"
For every inlinitely ditterentiable and nonnegative function ~/J(t,.v), the functional u - - , .j.'l'(Vu)2~/j(t,.v)d.vdt is convex and iower-semicontinuous on the weak space L 2 ((), /': HI), and thus
ff
)!!)~)
(Vu~)
2~/,(1, .,) d., dt >~ ff
( Vu
)2
1/, (t, ., ) d., d/.
which implies lim,_,~)i,(Vu, )" - ~,(Vu) 2 = D(u) ~> (). This fact is well known: see, for example, 136 ]. RI:,MARK. Two natural questions arise at this point: (I) Does there exist a weak solution of Navier-Stokes in the space L2((), 7": H i ) A L'V(0, T: L") with D ( u ) g: ()'? (2) Does the condition D(u) ~> 0 imply uniqueness for weak solutions of NavierStokes? Let us call such weak solutions with D(u) ~> ()"dissipative". in the case of inviscid Burgers equation in one-space dimension, D(u) ~> () coincides with the usual entropy condition of negative jumps, which does imply uniqueness. The fl)llowing proposition shows that the condition D(u) ~> 0 appears naturally fl)r weak solutions of Euler equation.
<
I~
7t.
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~ --.
,--'.
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~
~
~,
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<
0
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II
9
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'=
.
9- -
_
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.
,-,
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m"
+
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9
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. . . .
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r,
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tu
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50
R. Robert
integrating in ~ over the ball
O
m
I~1 ~ ~ one gets
'(s
3 im16rr }+()e
(u(x + sse) - u ( x ) ) 2 ( u ( x
+ s~i) - u ( x ) ) , se dZ'(se)).
i=l
Our expression of (u(x + s.se) - u ( x ) ) 2 ( u ( x
s(u) -- lim ~:-+() 8
+ eeq) _ u(.,c)), qedX'(se)
l= I
thus simply gives a local nonrandom form of the above expression of the inertial dissipation.
4. Last comments and acknowledgements We have tried here to give an overview of some topics which we feel inspired by Onsager's views. We l'ocussed as far its possible on issues which yielded rather precise mathematical developments. Doing so we w~luntarily discarded other important issues in turbulence theory such its intcrmittency or the recent important disc~vcry of the inverse energy cascade in 2D turbulence. it in a great pleasure for me to take the ()pportunity to warmly thank those wh(~ have had a direct contribution in the elaboration of the present work" F. Bouchet, P.H. Chavanis, J. Duchon (who greatly helped me to write Section 3. ! ), T. Dtlmont, J. Michel, A. Mikelic, C. Rosier, J. Sommeria. Great thanks also t(7 C. Bardos, B. Castaing, P. Constantin, A.J. Chorin, G. Eyink, M. Farge, U. Frisch, J.L. Lebovitz, A. Majda, R. Moreitu, who kindly contributed t() the discussion and diffusion of these ideas.
Appendix A
Ftitile-elellseitt ai~pro.vi, taliolt. F()r the comfort of the reader, we briefly recall some standard notations and properties 141. Apl~ro.vi, tatiolz o/ tile Sohole~' space H'"(IR a). We denote Q,/ - l - I/2, I/21 '1, Z the characteristic function of Q,/. and fi - X , ' " , X (,z + I terms). For a given parameter h > (), we define a prolongation operator I'/," to any function ./), -- y-~.i ./)ii X (7; .j) (.j beX
longs to Z '/), we associate the function
p,,.i>,- .li[t
(x) T-.J.
__
~
,..
.5
:>
9~
~
.
~
.>-
.~
,..a
.
9
~.
....
.....
...
<
.----- .~.
~'-*>
~
...~
E
to
~g
~>~>~>
.
-:~"~ > ~ . < ~ <
,..,.
.~
~I
-
---
a:
.<
/A
-4
//'x
._.
~
"-
~
r
----
---
-
.~
'.-2
~
7"
,...
P-,...,.
;>
.....
II
v....
,_0
~
"" ~ "
Z..
~ ~-.
,..,.<'6
-
~"
~
~,"
=a
~'~
~
2
r~
"*
~
"
---
/A
a:
9'
/A
"~
o
,r
~
:a-"
~
"a
"~~
9
.<
o~--
~
~
~-
~,,,
,ca
~
~
{'D
=
II
--" 9
9
m
,r
-"
~
q" ~
I
"
A
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~
II
", =
~~
o
=:
'-<
52
R. Robert
[4] J.E Aubin, Approximation of Elliptic Boundao,-Value Problents, Wiley-lnterscience, New York (1972). [5 ] E Baldi, l_xlrge deviations and stochastic homogenization, Ann. Mat. Pura Appl. 4 ( 151 ) (1988), 16 I - 177. [6] C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in 3 space wtriables with small initial data, Ann. Inst. H. Poincar6, Anal. Non Lin6aire 2 (1985), 101-118. [7] C. Basdevant and R. Sadourny, Modt;lisation des t;chelles virtuelles dans la simulation numdriqlte des t;t'oulements turbulents bidimensionnels, J. M6ca. Th6or. Appl. (Num6ro sp6cial) (1983), 243-269. [8] G. Benfatto, P. Pico and M. Pulvirenti, J. Statist. Phys. 46 (1987), 729. [9] J. Bertoin, The im,iscid Burgers equation with Brownian initial velociO', Comm. Math. Phys. 193 (2) (I 998), 397--406. [10] J. Bertoin, Lt;vv Processes, Cambridge University Press, Cambridge (1996). [l l] J.J. Binney and S.D. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, NJ (1987). [12] C. Boldrighini and S. Frigio, Eqltilibrium states for a plane incompressible perfect.fluid, Comm. Math. Phys. 72 (1980), 55-76. 113] E. Caglioti, EL. Lions, C. Marchioro and M. Pulvirenti, A ,v~ecial class of stationary .flows for twodimensional Euler equatirms: A statistical mechani('s description, Comm. Math. Phys. 143 (1992), 501525. [I 4] L. Carraro and J. Duchon, Solutions statistiques intrinsOqttes de i'~;qttatiott de Burgers et processus de Levv, C. R. Acad. Sci. Paris Ser. ! 319 (1994), 855-858. [I 5] l,. Carraro and J. Duchon, Equation de Burgers ave~" ~'onditions initiales gt a~'~'rois.s'ements ind~;pen~htnts el /tonloqOnes, Ann. Inst. H. Poincar6 15 (4)(1998), 431-458. 16 S. Chandrasekhar, Princo~les o.['Stellar Dvnantics, Dover, New York (1942). 17 P.H. Chavanis, J. Sommcria and R. Robert, Statisti~'al nle~'hani~'s o./lwo-dimen.s'iottal ~'orti~'e,s and ~'oilisionless stellar svstem.s, Astrophys. J. 471 (1996), 385-399. 18 A. Chorin, Statisti~'al Me~'hani~'.s and Vortex Motion, l,ectures in Appl. Math., Vol. 28, Amer. Math. Sot., Pr~vidence, R! (1991), 85. 19 P. Constal]ti!1, W. E and E.S. Till, ()ll.S'~Iq'CJ",S' ('hah's d'~qttation.s ' du t37~e Vla.~o~' l~oi.~.sott, ('. R. Acad. Sci. Paris 307 (1988), 655-658. 21 J. l)uchon and R. Robert, Inertial ellet~y dissilmtion.for weak sohttiotts Q/ill~'onqJre.ssihle Etder atui Na~'ierStokes equations, Nonlinearity 13 (2()()()), 249-255. 22 G. l-yink, Etterqy dissO~atiotl withottt i'isr in ideal hvdrodvttamir'x, l. l"ottrier atta/v.~is and Io~'al enet:~.v tran.sfer, Phys. I) 78 (3-4) (1994), 222-24(). 23 G.I,. Eyink and H. Spohn, Ne~atil'e states and larq,e-scale Ion,~-Ii~'ed i'orti~'es in l~l'o-dimensional ltlri~ltlent'e, J. Slalist. Plays. 70 (i 993), 833-886. 24 ('. t-oias, O. Manley, R. R()sa and R. Temam, Na~'ier-Stokes Eq/u/lion.s and 7"twlmh'm'(', Cambridge University Press, Cambridge (2()() I ). 25] U. Frisch, 7"ttrhtth'n~'e, Can]bridge University Press, Cambrigde (1995). 26] M. Hem)n, An. Astrophys. 27 (1964), 83. 27] T. Hida, Statiomtrv Sto~'hasti~" l~ro~'e.sses, Princeton University Press, Princeton, NJ (197()). 28l E. Horst and R. Hunze, Weak sohttions ~?/'the initial value prohlent.for the tmmodi/ied mmlitwar Vlasov equation, Math. Meth. Appl. Sci. 6(1984), 262-279. [29] E.T. Jaynes, 7"he Mittinmnt Etttrr Produ~'tiott Pritu'o~le, Collected Papers, R.I). Rosenkrantz, ed., Pallas Paperback Series, Kiuwer Academic, Dordrecht (1989). [3()] R. Jordan, A statisti~'al equilihrittnl model ~?/cohetvnt strttcturex in nta~netohydrodynanlics, Nonlinearity 8 (1995), 585-614. 131] l.R. King, Astrophys. J. 71(1966),64. [32] R.H. Kraichnan, Di['[itsion i~v a random ~'elocitv[iehl, Phys. Fluids 13(1970), 22-31. 133] R. Kubo, Sto~'hasti~" Liouville equation, J. Math. Phys. 4 (1963), 174-183. i341 J. Leray, l;,'tmle de diverses ~;qttatiotl.s int~;~rales tlotllitl~;aitvs el de qtu'lqtu's i~roblOnles qtte pose l'hydrodynantique, J. Math. Pures Appl. 12 (1933), 1-82. [35] J. Leray, Essai sur le mottventent d'un liquide visquettx entl~lissattt I'e.V~ace, Acta Math. 63 (1934), 193248.
Statistical hydrodynamics (Onsager revisited)
53
[36] EL. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Clarendon Press, Oxford (1996). [37] P.L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math. 105 (1991), 415-430. [38] D. Lynden-Bell, Statistical mechanics of violent relaxation in stellar systems, Monthly Notices Roy. Astronom. Soc. 136 (1967), 101-121. [39] C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, SpringerVerlag, New York (1994). [40] J. Michel and R. Robert, Statistical mechanical theory ~ t h e great red spot of Jupiter, J. Statist. Phys. 77 (3/4) (1994), 645-666. [41] J. Michel and R. Robert, Large deviations.for Young measures and statistical mechanics of infinite dimensional dynamical systems with conservation law, Comm. Math. Phys. 159 (1994), 195-215. [42] A. Mikelic and R. Robert, On the equation describing a relaxation toward a statistical equilibrium state in the two-dimensional perfect fluid dynamics, SIAM J. Math. Anal. 29 (5) (1998), 1238-1255. [43] J. Miller, Statistical mechanics of Euler equations in two dimensions, Phys. Rev. Lett. 65 (17) (1990), 2137-2140. [44] J. Miller, P.B. Weichman and M.C. Cross, Statistical mechanics, Euler equations, and Jupiter's red spot, Phys. Rev. A 45 (1992), 2328-2359. [45] D. Montgomery and G. Joyce, Statistical mechanics of negative temperature states, Phys. Fluids 17 (1974), I139-1145. [46] L. Onsager, Statistical hydrodynamics, Nuovo Cimento Suppl. 6 (1949), 279. [47] T. Padmanabhan, Statistical mechanics of gravitating systems, Phys. Reports 188 (5) (1990), 285-362. [48] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions .for general initial ~hmt, J. Differential Equations 95 (1992), 28 !-303. 149] R. Robert, A ma.vimttm entropy l~rim'iple.for two-dimensional Euler equations, J. Statist. Phys. 65 (3/4) ( 1991 ), 531-553. 1501 R. Robert, Uni~'itd de la solution faible ?t stq~port compact de I'dquation ~h" Vlasov-Poisson, C. R. Acad. Sci. Paris S6r. 1 324 (1997), 873-877. 1511 R. Robert, On the gravitational collapse of stellar systems, Classical Quantum Gravity 15 (1998), 38273840. 1521 R. Robert, On the statisti~,al me~'hani~'s of2D incompressible Euler eqttation, Comm. Math. Phys. 212 (I) (2000), 245-256. 1531 R. Robert and C. Rosier, On the modelling of small scales for 21) turbuh'nt flows, J. Statist. Phys. 86 (3/4) (1997). 1541 R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional.flows, J. Fluid Mech. 229 (1991), 291-310. [551 R. Robert and J. Sommeria, Relaxation towards a statistical equilibrium state in two-dimensional perfect .fluid dynamics, Phys. Rev. Lett. 69 (1992), 2276-2279. 1561 R. Sadourny, Turbtdent di[fusion in large scale.flows, Large-Scale Transport Processes in Oceans and Atmosphere, Willebrand and Anderson, eds, Reidel, Dordrecht (1986). 1571 J. Schaeffer, Global ~:vistence for the Vlasov-Poisson system with nearly symmetric data, J. Differential Equations 69 (1987), 111-148. 1581 D. Serre, SvstPmes de lois de conservation, I. Diderot, ed. (1996). 1591 Z.S. She, E. Aurell and U. Frisch, 77w inviscid Burgers equation with initial data of Brownian type, Comm. Math. Phys. 148 (1992), 623-64 !. 1601 V. Scheffer, An inviscid.flow with compact support in space-time, J. Geom. Anal. 3 (4) (I 993), 343-401. 1611 A.I. Shnirelman, Lattice theory amlflows of ideal incompressible fluid, Russian J. Math. Phys. 1 (I) (1993), 105-114. 1621 A.I. Shnirelman, Weak solutions ~['incoml~ressible Euler equations with decreasing energy, S6minaire EDP Ecole Polytechnique Expos6 16 (1996-1997). [63] Ya.G. Sinai, The statistics of shocks in the solutions of inviscid Burgetw equation, Comm. Math. Phys. 148 (3) (1992), 601-621. [64] J. Sommeria, C. Note, T. Dumont and R. Robert, Th6orie statistique de la tache rouge de Jupiter, C. R. Acad. Sci. Paris S6r. II 312 (1991), 999-1005.
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[65] J. Sommeria, C. Staquet and R. Robert, Final equilibrium state of a two-dimensional shear layer, J. Fluid Mech. 233 ( 1991 ), 661-689. [66] A. Thess, J. Sommeria and B. Jtittner, Inertial organiz.ation of a two-dimensional turbulent vortex street, Phys. Fluids 6 (7) (1994), 2417-2429. [67] B. Turkington and R. Jordan, Ideal magnet~uid in two dimensions, J. Statist. Phys. 87 (3-4) (1997), 661695. [68] G.J.F. Van Heijst and J.B. Flor, Dipole formation and collision in a stratified fluid, Nature 340 (1989), 212-215. [69] S.R.S. Varadhan, Large deviations and applications, Ecole d'l~t6 de Probabilit6s de Saint-Flour 15-17, i 985-1987. [70] V.I. Yudovich, Non-stationary flow of an incompressible liquid, Zh. Vych. Mat. 3 (1963), 1032-1066. [71] L.C. Young, Generalized surfaces in the calculus of variations, Ann. Math. 43 (1942), 84-103. [72] C.K. Zachos, Hamiltonian flows, SU(oo), SO(oo), USp(oo) and strings, Differential Geometric Methods in Theoretical Physics, L.L. Chau and W. Nahm, eds, Plenum Press, New York (1990). [73] V. Zeitlin, Finite mode analogs of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure, Phys. D 49 ( 1991 ), 353-362.
1
Statistical Hydrodynamics (Onsager Revisited) Raoul Robert Institut Fourier CNRS, Universit6 Grenoble 1, UFR de math6matiques, BP 74, 38402 Saint Martin d'Hbres cedex, France E-mail: raoul, robert@ ujf-grenoble.fr
Contents I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Equilibrium problems: Self-organization of the turbulent flow . . . . . . . . . . . . . . . . . . . . . . . 2.1. Equilibrium states for 2D incompressible ideal flows . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Extension of the theory to other systems, the Vlasov-Poisson equation . . . . . . . . . . . . . . . 2.3. Relaxation towards the equilibrium and parametrization of the small scales . . . . . . . . . . . . . 3. Out-of-equilibrium problems: Weak solutions, shocks, and energy dissipation . . . . . . . . . . . . . . 3. I. Statistical solutions of I D inviscid Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Energy dissipation for 3D flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Last comments and acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H A N D B O O K OF M A T H E M A T I C A L FLUID DYNAMICS, V O L U M E II Edited by S.J. Friedlander and D. Serre 9 2003 Elsevier Science B.V. All rights reserved
3 4 6 25 31 37 37 44 50 50 5i
Statistical hydrodynamics (Onsager revisited)
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1. Introduction
Though it would be soundly enlightening to discuss thoroughly how statistical methods were introduced into the field of turbulence, we intend to limit ourself to only a few preliminary comments. In its crudest form the statistical approach consists in splitting the turbulent velocity field into a smoothly varying part plus a random fluctuation. This naturally yields the notion of turbulent viscosity (Reynolds, Boussinesq). From the beginning of the last century, systematic efforts were made by prominent scientists to elucidate the enigma of turbulence by statistical methods. In most cases these efforts amount to perform more or less formal calculations on random solutions of Navier-Stokes equation, solutions which are supposed to exist, provide a relevant description of turbulent flows and moreover satisfy some symmetry properties. The main success of this approach is the famous 4/5 law of Kolmogorov, Von Karm~.n and Howarth which gives a precise relationship between the energy dissipation and the third moment of the velocity increments (see Frisch [25]). More recently further progress was made on the mathematical aspect of this statistical approach when two important issues were addressed: (l) Give a precise definition of statistical solutions. (2) Prove existence results for such solutions. For these aspects we refer to [24]. Our purpose here is to expound some developments of the statistical approach which are directly inspired by the original ideas of Onsager on turbulence [46]. Onsager's starting point is a clear distinction between the two and three-dimensional cases: in two dimensions the energy of the turbulent flow is conserved, and the main phenomenon to explain is the self-organization of the flow into coherent structures (large scale eddies). On the other hand, in three dimensions, the dissipation of the energy persists in the limit of vanishing viscosity and we observe a power law energy spectrum. The distinction made by Onsager is brilliantly shown by the experiments of Van Heijst [68]. In the first part we will try to carry through Onsager's program on 2D turbulence, that is, try to extend in a rigorous way to hydrodynamics the statistical mechanics approach of Boltzmann. We will also indicate some practical consequences concerning numerical simulations. In the second part we consider the 3D case. Here also we follow Onsager's ideas: we do not start considering solutions of Navier-Stokes equation and then make the viscosity go to zero (what we may call Leray's point of view) but we suppose that the turbulent flow might be correctly described by weak solutions of Euler equation which are not regular enough to conserve the energy. Despite the fact that the Cauchy problem for such solutions is still unsolved, examples of such weak solutions with energy dissipation have been recently constructed. We begin by considering the simpler case of the one-dimensional Burgers equation which retains some of the interesting features of our problem (energy dissipation for weak solutions with shocks). Due to the discontinuities in the solution the energy will decrease. It is natural to try a statistical description of the dynamics of these shocks, that is, to describe the action of Burgers flow on a stochastic process. A first answer to this issue appears astonishingly simple: the class of Levy's process with negative jumps is preserved by Burgers equation.
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R. Robert
Of course, it is tempting to imagine a similar approach for Euler equation. Obviously, things are much more difficult. As a first step we propose a formula giving the expression of local energy dissipation due to the lack of regularity of weak solutions. This leads to a natural entropy condition for such solutions.
2. Equilibrium problems: Self-organization of the turbulent flow The most striking feature of 2D hydrodynamical turbulence is the emergence of a largescale organization of the flow, leading to structures usually called coherent structures (see references in [17,54]). Jupiter's Great Red Spot, a huge vortex persisting for more than three centuries in the turbulent shear between two zonal jets, is probably related to this general property [40,64]. Such hydrodynamical vortices, whose dynamics is governed by Euler equation or some quasi-geostrophic variant, occur in a wide variety of geophysical phenomena and their robustness demands a general understanding. Similarly, the galaxies themselves follow a kind of organization revealed in the Hubble classification [11]. The dynamics of Galaxies is dominated by stars under collective gravitational interaction rather than gas or hydrodynamical processes. In particular, for most stellar systems the collisions (i.e., close encounters) between stars are quite negligible and the galaxy dynamics is well modeled by the Vlasov-Poisson equation. The common remarkable feature of these structures is that they occur and persist in a strongly turbulent environment. In the case of 2D turbulence, Onsager [46] was the first to suggest that an explanation might be found in terms of statistical mechanics of Euler equation. In the case of galaxies, the natural approach, which consists in defining a statistical equilibrium for a cloud of stars, fails (see references in [17]); this is mainly due to the fact that the relaxation time associated with the "collisions" of stars widely exceeds the age of the universe. This lead astrophysicists to suggest that it is a much more efficient mixing process generated by the collisionless Vlasov-Poisson equation which drive the system towards some sort of equilibrium. In [ 17] we stress, at a physical level, the analogy between the violent relaxation of stellar systems and the mixing of vorticity yielding coherent structures in 2D turbulence. This analogy resides in the similar morphology of the Euler and Vlasov equations, and we try to explain their self-organization with the same theoretical tools. Let us now be more precise. In his pioneering paper, Onsager's argument on the possibility of negative temperature equilibrium states was based on the approximation of the continuous Euler system by a great (but finite) number of point vortices. This leads to a finite-dimensional Hamiltonian system, to which the methods of statistical mechanics can be applied (see, for example, [13,45]). Though qualitatively very enlightening, this approach reveals severe difficulties; for example, there are many different ways to approximate a continuous vorticity by a cloud of point vortices and different approximations can lead to very different statistical equilibrium states, so the thermodynamical equilibrium that we can associate to a continuous vorticity depends dramatically on arbitrary choices (this difficulty was underlined by Onsager). Thus, if we want to produce reliable quantitative predictions on the actual behavior of the flow we have to proceed differently.
Statistical hydrodynamics (Onsager revisited)
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A natural way to define equilibrium states is to construct invariant Gibbs measures on the phase space. But we do not know how to construct such measures on the natural phase space L ~ (Y2) for Euler equation. Some work has been devoted to the study of Gibbs measures with formal densities given by the enstrophy and the energy [8], and also to Gibbs measures associated with the law of vorticity conservation along the trajectories of the fluid particles [ 12]. Unfortunately, all these measures are supported by "large" functional spaces so that not only the mean energy and enstrophy of these states are infinite but the phase space L ~ ( s is of null measure. So, it is only at a formal level that this makes sense. Moreover this approach fails to give any prediction on the long time dynamics corresponding to a given initial vorticity function. The most common approach to overcome these difficulties is to use a convenient finitedimensional approximation of the system, possessing an invariant Liouville measure. Then one can consider the canonical measures associated with the constants of the motion and try to perform a thermodynamic limit in the space of generalized functions when the number of degrees of freedom goes to infinity. For example, for Euler equations one can consider the N Fourier-mode approximation or the point-vortex approximation. Two difficulties arise in this approach. The first is to choose a relevant scaling to perform the limit, the second is even more fundamental: generally, the approximate system will have less constants of the motion than the continuous one, so that the long-time dynamics of that system may be very different from that of the continuous one. For more comments and references on these attempts see, for example, [23,49,54]. To overcome the difficulties evoked above, our approach is based on the following points. 9 We work on an extended phase space (the space of Young measures) on which the constants of the motion put natural constraints. 9 We construct a sequence of finite-dimensional approximations of the Euler flow (with good convergence properties such as strong L 2 uniform convergence on any finite time interval), satisfying the two tbllowing properties: (i) A Liouville theorem holds for the finite-dimensional approximations. (ii) For the family of measures given by (i), we can prove the Sanov-type large deviation estimates for empirical Young measures which are necessary to take the thermodynamic limit [41,49,52]. Of course, it is easy to satisfy the point (i) by considering the spectral approximation; but then, it is a very difficult issue to prove that the associated family of measures satisfies (ii). Our approach here is to get a Liouville theorem for a general class of approximations, including approximations on spaces of functions which are spatially localized like finite element approximants. For such approximations we are not able to prove directly the large deviation estimates (ii) but we can use them as an intermediate to construct the final approximation on the space of piecewise constant functions for which the large deviation estimates hold [41]; so that the use of the finite-element approximants appears here as an essential intermediate step in order to both insure the convergence of the approximations and keep the large deviation estimates. One might worry about the fact that our approximate dynamical system retains only the enstrophy among the infinite family of the Casimir functionals which are conserved by the continuous system (in contrast with the finite mode Hamiltonian approximation
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R. Robert
of [72,73]). Of course, it would be more satisfactory to construct approximations having in addition a large number of constants of the motion. But we think that this is not truly necessary to our microcanonical approach. Indeed, if we are interested in the long-time behavior of 2D-Euler flow, and if we believe that a statistical mechanics approach can bring some light on this issue, then we expect that we will finally have to solve some constrained variational problem: find the maximum value of some entropy functional under a set of constraints. But while we have no doubts about the set of constraints which is directly derived from the constants of the motion of the system (energy, integrals of functions of the vorticity field ... ), it is hard to guess what the relevant entropy functional is. But in our microcanonical approach the entropy is not related to the fact that many constants of the motion are (or are not) exactly conserved by the approximate flow but it is only associated to large deviation estimates for the invariant measures.
2.1. Equilibrium states for 2D incompressible ideai flows 2.1.1. 2D Euler equation. The motion of a two-dimensional incompressible inviscid fluid in a bounded domain s is governed by Euler equation, which we write in the classical velocity-vorticity formulation:
(E)
{wl + div(wu) - 0, curlu--(o, divu--0,
u.n=O
on~)s
where u(t, x) is the velocity field of the fluid, ~o = curl u the scalar vorticity, n the outward unit normal vector to ~)s Because of incompressibility we introduce the stream function
O(t, x): ~o=-/xg,,
~=0
on 0,(2.
The constants of the motion of this dynamical system are: - the energy
,fu
,T, ( w ) - ~
dx-
~
~codx;
- the integrals
F0(~o)- fa O(o)(x))dx, for any continuous function 0. These constants of the motion which are associated to the degeneracy of the (infinite-dimensional) Hamiltonian system are usually called Casimir functionals.
Statistical hydrodynamics (Onsager revisited) -
7
if #2 is the ball B(0, R), we must consider also the angular momentum with respect to 0: M(~) =
'(fo
x A u(x) dx = ~
(R 2 - x 2 ) ~ ( x ) dx
)
e3.
The Cauchy problem. Yudovich's theorem [70] gives a satisfactory existence-uniqueness result for the Cauchy problem for (E): For any given initial datum co0(x) in the space L ~ ( ~ ) , there is a unique weak solution of (E); this solution co(t, x) is in L ~ ( ~ ) for all t, and furthermore belongs to the space C([0, ~ [ ; LP(ff2)) for all p, l~
The turbulent mixing process. Let us now briefly describe the mechanism of turbulent mixing which is responsible for the self-organization of the flow in Euler equation. As we have seen, Euler equation can be described as the advection of a scalar function (the vorticity) by an incompressible velocity field with which it interacts via a Poisson equation. The vorticity is not passively advected by the flow but is coupled with its motion, this coupling will be responsible for the fluctuations of the stream function which will mix the vorticity at small scale and induce a self-organization and the appearance of structures at larger scales. This process is studied at a physical level in [17]. Our concern here is to introduce an entropy functional which will give a precise content to the vague notion of turbulent disorder of the flow. Of course, to define such a functional, following Boltzmann's approach, one would have to choose a relevant measure on the phase space. It is well known that, at a formal level, Euler equation is an infinite-dimensional Hamiltonian system; but, in contrast with the finite-dimensional case, this does not imply the existence of an invariant Liouville measure on the natural phase space L ~ . Fortunately, it occurs that to define the entropy functional we do not actually need to have a Liouville measure on the infinite-dimensional phase space, we only need the existence of finite-dimensional approximations which admit invariant Liouville measures, large deviation theory will then yield the relevant entropy. This is the very root of our thermodynamical approach. Although we can find finite-dimensional approximations of Euler equation which preserves the Hamiltonian structure [72,73], this structure is broken by any kind of approximation of practical use. But tbr the needs of thermodynamics the Hamiltonian structure is not truly necessary, it is the Liouville theorem and the constants of the motion which are the key ingredients. In the case of Euler equation, it is well known that a Liouville theorem holds for the usual spectral approximation. We shall show that this is a particular case of a general property: there is a natural way to approximate Euler equation on any finite-dimensional space in such a way that the volume measure is conserved. The spectral approximation is only a particular case of that. Then the problem of defining an equilibrium statistical
8
R. Robert
mechanics for (E) amounts to the study of families of measures. For an arbitrary choice of the approximating spaces the study of the asymptotic behavior of these measures seems untractable, but fortunately we can choose spaces for which the thermodynamic limit of these measures can be carried on [41,52]. 2.1.2. Finite-dimensional approximations and Liouville theorem. A classical way to construct finite-dimensional approximations of Euler equation is as follows. Let FN be an N-dimensional subspace of L ~ ( ~ ) and denote by PN the orthogonal projector from Lz(~Q) onto FN. Then we define the approximate solution coN(t) as the solution of the ordinary differential equation in FN:
(EN)
where
U N -- curl
coN + PN(U N. Vco N) --O, (.0N (0) -- PN COO
1/./"N,a n d
- - A 1 / f N - - coN, 1//,N _ 0 o n 0 F 2 .
If FN is properly chosen and w0 is regular enough, then (_,ON (t) converges towards co(t) for the strong L 2 topology, uniformly on any bounded time interval [39]. The constants of the motion of the dynamical system (EN) are: - the energy 89fs2 1//'U CONdx, - the enstrophy J~2 (CON)2 dx. Let us notice here that (EN) is a differential system with quadratic nonlinearity so that the solution always exists on a small time interval; but due to the conservation of the enstrophy the solution cannot blow up and it exists globally in time. Now, it is well known that if we take for FN a subspace generated by N eigenvectors of the operator - A (with the Dirichlet boundary condition), the volume measure on FN is conserved by (EN). This is in fact a particular case of what follows. We consider the modification of ( E N ) which consists in replacing, in the definition of ~ U the Dirichlet problem by the variational formulation:
g/NEFNand
fs27~tN.7~odx=fs2coN~odx,
for all r in FN.
For the sake of simplicity, from now on we shall also denote by (EN) this modified dynamical system. Of course, we shall suppose at least that FN is included in the Sobolev space Hr ( ~ ) , so that for any given w N, the above variational problem possesses a unique
solution 1/J u (by the Lax-Milgram theorem). One can easily check that the energy and the enstrophy are still conserved but now we have in addition THEOREM 2. I.I. The volume measure on FN is conserved by the dynamical system (EN). PROOF. FN is endowed with the L 2 scalar product. Let us write (EN) in the form coN = G N (CON), where G N(cou ) -- -- PN ( uN .TcoU) is a nonlinear transformation of FN. Then to
Statistical hydrodynamics (Onsager revisited)
9
prove the theorem it suffices to show that the trace of the d e r i v a t i v e GiN(coN) vanishes. Let us compute the first variation of GN corresponding to a small variation ~coN. -
o;,
] -
-
-
By definition, we have tr(G~v (coN)) -- ~--~i(G%(coN)[ei], ei), for any orthonormal basis ei of FN. Let us denote ui the vector field associated to ei, we have:
(G'N(coN)[ei],ei)---- fs2(ui'VcoN)eidx-- fs2(uN'Vei)eidx, but since div U N "~ 0, the last term vanishes, and after integration by parts we get:
(G~N(coN)[ei], el) -- f ~ coN curl ~i " Vei dx. Let us consider now the positive definite and symmetric linear operator A defined on FN by: fs2 V7 r" X7qgdx = (ATr, qg), and take for ei an orthonormal basis of eigenvectors of A. We obviously have ei -~ )~i~i (~.i is the eigenvalue corresponding to e i ) , so that curl ~ i . V e i -- 0 and tr(G~(cox)) -- 0. I-7 Now, it remains to prove the convergence (when N --+ (x~) of the approximate solution coN (t) towards the solution co(t) of the Euler equation.
We shall take for approximating space FN the space Fh (s of the finite-element approximation of the Sobolev space H m (R2), with compact support in s (m is an integer > 5 and h a small positive parameter, see Appendix A). Then we have the following convergence result whose proof is classical. PROPOSITION 2.1.2. Let co(t) be any weak solution of (E), with co0(x) in the .space
L~(S2), and let T > 0 be fixed. Then for all e > O, there is h(E) > O, such that for all h, 0 < h <~h(e), there is a solution cob(t) of (Eh) such that:
lifo(t)- oJ'(t)llLZ(X)
,larunt
in
10, TI.
As we will see later, the measures /Zh on Fh(12) associated with this approximation are not easy to handle, but it appears that a slight change in the approximating dynamical system improves greatly the situation with a view to our thermodynamical purpose. Let us denote by Qh the flow on Fh (S2) defined by the system (Eh). Let Ph "LI21~ Fh be the classical prolongation operator of the finite-element method (see Appendix A), and -I Yrh ~- Ph "
Let us define Lh (s -- n'h Fh (s and denote 6')~ - 7rh o Qh o Ph, the flow induced on Lh(S2). Obviously, O)/~ preserves the volume measure on Lh (S-2). From Proposition 2.1.2 we deduce the following.
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R. Robert
COROLLARY 2.1.3. Let co(t) be any weak solution of (E), with co0(x) in the space Lee(F2), and let T > 0 be fixed. Then f o r all s > O, there is h(s) > 0 such that f o r all h, 0 < h ~ h(s), there is coOh in Lh(F2) such that:
II oc, - O, OOh II
for all t in [0, T].
PROOF. By the L2-stability property of Euler equation, we only need to prove the result for coo in C ~ (Y2). Using Proposition 2.1.2, we have, for h <~ h(s): Ilco(t) - coh(t)]l ~< s, on [0, T]. Let us denote c o b ( t ) = n-hcoh(t), we have: I l c o ( t ) - coh(t)ll ~< I l c o ( t ) - rhco(t)ll + Ilrhco(t) - coh (t)II, where rh is the classical restriction operator (see Appendix A). But since
it becomes:
Now we have (see Appendix A)
chll o(')ll
C(T)h,
on [0, T],
and similarly
thus lifo(t) - OJh(t)ll ~ C ( T ) h + cs, and the result follows.
73
Let us summarize our results. We have constructed a flow (,-)]' o n Lh(~2) which approximates the Euler flow and preserves the measure dcoh - - ( ~ ) j dcoil, where coh(X) -J (x / h - . j ) (finite sum). y~j cohX 2.1.3. Baldi's large deviation theorem and thermodynamic limits. In order to define relevant statistical equilibrium states, we have to take the thermodynamic limit of the invariant Liouville measures with the conditioning given by all the constants of the motion. To perform this task we need some tools from large deviation theory. Baldi's theorem gives general conditions under which a family of probability measures on a locally convex topological vector space has the large deviation property. As we will see, it provides a powerful tool to carry out thermodynamic limits for infinitedimensional systems.
Statistical hydrodynamics (Onsager revisited)
11
The large deviation property. Let E be a locally convex Hausdorff topological vector space. We consider a family #h, h > 0, of Borel probability measures on E. We will say (see, for example, Varadhan [69]) that the family #h has the large deviation property with constants )~(h) and rate function L iff: (i))~(h) is > 0 and limh--->+~ ~(h) = +oo. (ii) L : E --> [0, +o0] is a lower semi-continuous functional on E (not identical to +oo). Moreover, L is inf-compact, that is: the set {vl L ( v ) <. b} is compact for all real numbers b. (iii) For every Borel subset A of E, we have:
- A ( A ) ~< l i m i n f ~ l log#h(Z), h - ~ ~.(h) and 1
lim sup
LogXh (A) ~ < - A ( A - )
h--->o~ ~
where A ( A ) --inf,,cA L(v). The functional L is also usually called the information functional, and - L the entropy functional. Let E' be the topological dual of E, endowed with the weak-star topology cr (E', E). For a Borel probability measure/z on E, we define its Laplace transform: fi(cP) -- ft; exp({cp, v ) ) d # ( v ) ,
for r 6 E'.
As it is well known,/~ is a convex, lower semicontinuous and proper functional on E'. The same is true for the functional Log/~(r BALDI'S THEOREM 2.1.4. Let ~h be a family o f Borel probability measures on E, satis.~ing the following assumptions: (1) There is a function )~(h) as in (i) such that 1
lim Log ~h (~.(h)q)) -- F(cp), h ~ ~.(h) where F is a convex, lower semicontinuous and proper functional on E' which is finite on a neighborhood o f the origin. (2) Compacity assumption: For every. R > O, there is a compact set K R C E such that 1
lim sup ~ Log #h ( K " ) <~ -- R. h--,~ ~(h) R. Let us denote by L the Young-Fenchel transform o f F, that is: L ( v ) = sup ({r r E'
v)-
F(cp)),
.for v ~ E.
12
R. Robert L is a convex, lower semicontinuous, and proper functional on E. Baldi's theorem states that under the assumptions (1) and (2) the upper bound in (iii) holds. If we suppose that L has some additional strict-convexity property, we can also derive the lower bound. We will suppose that L satisfies the following condition. (3) For every real number r, the set Ar = {v I L ( v ) <~ r} is the closure o f the subset o f the points v o f Ar where the subdifferential 0 L ( v ) is nonempty and contains an element %osuch that:
L(u') > L(u) + (tp, u ' - u),
for all u' ~ u.
Then Baldi's theorem asserts that under the hypotheses (l), (2), (3) the two bounds in (iii) hold. So, we see that the family ~h has the large deviation property with constants ~.(h) and rate function L. Indeed, one easily checks that the functional L is inf-compact on E: for every real number b , the set A/, is closed and the lower bound applied to the open set K 'b+l yields At, C K/,+l (with the notation of (2)).
Comments. (1) In practice it may be difficult to check that the hypothesis (3) is satisfied. In fact, Baldi's proof works as well with the following weaker hypothesis (3'). (3') For every v such that L(v) < + o o , for every open set O containing v and every s > 0, there is vl c O such that L ( v l ) <<. L ( v ) + s and L is strictly convex at vl, that is: ----]~pc i)L(vl) such that
L(,,') >
,,' #,,,.
(2) L is strictly convex at v if, fl)r example, i)L(v) is nonempty and
for a l l 0 < t < I , v ' 6 d o m L , v':~v. (3) In the case where only the hypotheses (l), (2) are satisfied, as we have seen, Baldi's theorem gives an upper bound. But the functional L may fail to be inf-compact in that case. Nevertheless, we can see that the set A0 -- { v E E I L (v) -- 0}
is nonempty.
Notice first that we obviously have F ( 0 ) = 0 and since F is also the Y o u n g - F e n c h e l transform of L, we get: inf L(v) = O.
vEE
Furthermore, we have g . i ( K i ) + l t j ( K ' l ) -- 1 and from (2) we know that l~h(K'l) --+ 0 (when h --+ oo).
Statistical hydrodynamics (Onsager revisited)
13
Now, if A0 were empty we should have A (Kl) > 0. Then, applying Baldi's theorem we should have (/Zh (K1) ~ 0); this would yield a contradiction. Moreover, one can easily deduce that for any open set U containing A0 there is a number ot > 0 such that:
#h(U':) <~exp(-X(h)c~),
for h large enough.
We shall say that the family #h concentrates about the set A0.
Thermodynamic limits and the concentration property. When dealing with thermodynamic limits one usually encounters the following situation, which we resume here in an abstract form. Let 3h be a family of random variables with values in a Hausdorff locally convex topological vector space E. 6h generally comes from some finite-dimensional approximation of an infinite-dimensional system. If we can prove that, for h large, with a high probability, 3h remains in a neighborhood of some points v* of E, then v* is the equilibrium state of our system, and the thermodynamic limit is performed. Large deviation theory shows that such a situation is very common. We will assume in the sequel that the family 6h (or the associated probability distributions/Zh on E) has the large deviation property with constants s and rate function L. Since Prob(gh ~ E) -- 1 for all h, we have inf,,~- L(v) -- 0 and the set A0 is a nonempty compact subset of E. And, as in the above comment 3, for every open set U containing A0, there is some c~ > 0 such that: Prob(ah 6 U") ~ e x p ( - 1 ( h ) o t ) ,
for large h.
That is, the family 3h concentrates about the set A0 which is the equilibrium set of the system. In our "microcanonical" approach, we will study now the situation where &l satisfies some given constraints (it would be more correct to say that we introduce some conditioning on the random variables 6h). These constraints will be given, for example, by the constants of the motion of an infinite-dimensional dynamical system. We introduce the constraints in the general form 3h 6 s where s is some subset of E. Of course, since 3h comes from a finite-dimensional approximation, the ideal constraints 6h 6 s will not be exactly satisfied, but only up to some approximation given by an open neighborhood of 0 in E, W. Let us denote s = s + W. We shall then consider 6h 6 s Let us now give a definition. DEFINITION. Let g, s be subsets of E. We will say that 6h concentrates about g* conditionally to g iff: I Log Prob(3h E s > -oo (i) 'v'W', lim infh--,oo z--~
(ii) 'v'W*, 3or > 0, 3W, VW', Prob(3h 6 s
\s
Prob(6h E Cw,)
)
~< e x p ( - 1 ( h ) o t ) ,
for h large enough.
14
R. Robert
Here W*, W, W I denote open neighborhoods of 0 in E. REMARKS.
(1) Heuristically, this definition means that if we know that 6h takes its values in a neighborhood of E, then it will be in a neighborhood of E* with a high probability. (2) As previously noticed, we have to widen the sets ,5', E* into open neighborhoods. In fact, Prob(3h 6,5') is not defined for an arbitrary subset C; and even if E is a Borel subset, it can be zero. (3) The condition (i) ensures that, when h ---> cx~, Prob(ah ~ Cw,) cannot be too small. Now we derive the following concentration result which will be useful to carry out thermodynamic limits. CONCENTRATION THEOREM 2. 1.5. We suppose that 6h has the large deviation property
with constants X(h) and rate function L. Let C be a nonempty closed subset of E and E* the subset of E where L achieves its minimum value on E. Then 6h concentrates about C* conditionally to C. PROOF. See [41 ].
1-3
REMARK. The set ~'*, in which 3h approximately remains with a high probability, is the equilibrium set of the system. If ~'* does not reduce to a point (the equilibrium state), we are in a phase transition situation. We shall see in the following how we can use this concentration result to derive a maximum-entropy principle for Young measures. 2.1.4. A maximum-entropy principle for Young measures. It currently happens when dealing with a limit process for a sequence of bounded measurable/'unctions that the sequence does not converge and shows an oscillating limit behavior, whereas some estimates and conservation laws hold. In such a case, the concept of Young measure has been found relevant to describe the behavior of the sequence (examples can be found in hyperbolic systems of conservation laws, homogenization, h y d r o d y n a m i c s . . . ). Young measures can be viewed as giving a macroscopic description of the system, whereas the bounded measurable functions are all the microscopic states. We use the results of Section 2 to derive a maximum entropy principle for Young measures, that is: the macrostate (Young measure) which realizes the maximum of an entropy functional has a natural concentration property (a large majority of the microstates satisfying a given set of constraints are in a neighborhood of that macrostate). It turns out that this entropy functional is the classical Kullback entropy (see Sanov's theorem).
Young measures. Throughout this section, X, Y will denote two locally compact separable and metrizable topological spaces. Let us suppose that a positive Borel measure dx is given on X. Let us recall that Young measures [71] are a natural way to generalize the notion of measurable mapping from X to Y: at any point x 6 X, we no longer have a well-determined value, but only some probability distribution on Y.
Statistical hydrodynamics (Onsager revisited)
15
In other words, a Young measure v is a measurable mapping x ~ Vx from X to the set Ml (Y) of the Borel probability measures on Y endowed with the narrow topology. Clearly, v defines a positive Borel measure on X x Y (that we will also denote by v) by:
(v, f ) -
fx(VX, f (x, . ))dx,
for every real function f ( x , y), continuous and compactly supported on X x Y ( f Cc(X x Y)). Moreover, for f ( x ) ~ Co(X), we have
(v, f ) - f x
f (x) dx,
that is, the projection of v on X is dx. It is well known [41 ] that this property gives an equivalent definition of Young measures. That is, for any positive Borel measure v on X x Y whose projection on X is dx, there is a measurable mapping x ~ Vx such that the above formula holds. The mapping x --+ Vx is unique up to the dx-almost, everywhere equality. To any measurable mapping f " X -+ Y, we associate the Young measure 6 f ' x --+ ~.f(x), Dirac mass at f (x). We shall make two additional assumptions" (,) The measure dx is diffuse and of finite total mass. (**) There is a distance function d(x, x') giving the topology of X, such that" for all s > 0, there is a finite partition of X into measurable subsets X = {X i [i -1. . . . . n(3~)} with I x i l - IXJl for all i, j (we shall say that 3~ is an equipartition of X), and satisfying d(,t~) ~< s, where d(3~) - sup/SUpx.x,~Xi d(x, x') is the diameter of Y. Notice that (,) and (**) imply that IA [ approaches zero when the diameter of a measurable set A approaches zero. Hypotheses (,), (**) are satisfied, for example, if X is an open convex and bounded subset of R" with dx -- Lebesgue's measure, and also if we consider any image of X by a dx-preserving homeomorphism. We shall denote by M the convex set of Young measures on X x Y, and we recall some useful properties. 9 M is closed in the space Mh(X x Y) of all bounded Randon measures on X x Y (with the narrow topology), the narrow topology is equal on M to the vague topology (weak topology associated with the continuous compactly supported functions) and it is metrizable. Furthermore, if Y is compact then M is compact. In the sequel M will be endowed with the narrow topology. 9 {~f[ f ' X ~ Y measurable} is a dense subset of M. This property is classical in the case Y is compact (see reference in [41 ]). The general case follows by approximation. Approximate first (for the vague topology) a given Young measure v by vx (as in the proof of Theorem 2.1.6 below) which is constant, equal to v i, on each set X i of an equipartition X, and then approximate each v i by a probability measure with compact support.
!6
R. Robert
A large deviation property. Suppose now that we are given a basic Borel probability measure n'0 on Y. Then with any equipartition 5E of X we can associate a Borel probability measure #:~ on M in the following way. We take Yl . . . . . )~n, n(3~) Y-valued independent random variables with the same distribution n'0. We consider the random function f.~ -- Z
y i 1xi, i
where 1xg is the characteristic function of the set X i . We denote by 3x the Young measure associated with f.~, and by # x the probability distribution on M of the random variable 6x. Now, we can state the main result of this section. THEOREM 2.1.6. When d(SE) ~ O, the family #:~ has the large deviation property with constants n ( Y ) / I X [ and rate function Ijr (v), where 7r - dx | 7ro and Ijr (v) is the classical Kullback information functional (see Varadhan [69]), defined on M by: Irr (v) --
{
f x ~ v Log ~dt, dr,
if v is absolutely continuous with respect to Jr, otherwise.
PROOF. The proof is an application of Baldi's theorem, see [41 ] for details.
[-I
In this Young measure framework, Theorem 2.1.5 yields the ff)llowing. COROLLARY 2.1.7. Let C be a nonempty closed subset of M, r the subset o[C where the.f'unctional l~r achieves its minimum value on C. Then 3x con~'entrates about E* conditionally to C. REMARKS. 9 Note that since the functional/Tr is inf-compact and ,5' is closed, ,5'* is nonempty. 9 Theorem 2.1.6 appears as a generalization of the well-known Sanov's theorem. Indeed, apply the contraction principle to the mapping v ---> f vx dx. 2.1.5. Thermodynamic limit of the invariant measures in the space of Young measures.
Long time dynamics and Young measures. As we have seen, Euler system describes the advection of a scalar function (the vorticity) by an incompressible velocity field, thus the vorticity oJ remains bounded in L~(,q2). The functionals
Co ( o 2 ) - ff2 O(~o)(x)dx, are constants of the motion (for any continuous function 0). That is to say, the distribution measure of co, trio, defined by (trio, 0) = Co (co), is conserved by the flow. Let us consider an initial datum coo. It is well known that, in general, as time evolves, F~o~0 becomes a very intricate oscillating function. Let us denote r = [[~o0i[L~(~). Since the
Statistical hydrodynamics (Onsager revisited)
17
measure n'o~ is conserved, Ftcoo will remain, for all time, in the ball Lr~ -- {co: I1~o11~~ r}. Extracting a subsequence (if necessary), we may suppose that, as time tends to infinity, Ftco0 converges weakly (for the weak-star topology o'(L ~ , L1)) towards some function co*"
Ftco0
w > 09, 9
We can easily see that Co(l-)coo) does not converge towards Co(w*) if 0 is nonlinear, whereas some other invariants can converge, as it is the case for the energy. So, much information (given by the constants of the motion) is lost in this limit process. Thus, the weak space L ~ (f2) is not the good one to describe the long-time limits of our system. Fortunately, the relevant space to do this is well known, it is the space Mr of Young measures on f2 x I - r , r], that we have just defined. We can identify the long-time limits of the system as Young measures. Indeed, Mr is a suitable compactification of Lr~ since the narrow convergence (when t approaches infinity) of 6~oo towards some Young measure v preserves the information given by the constants of the motion, that is, for all functions 0(z):
fs 0(r,~o()(x))dx -+ fa <~x,e>dx, but the left-hand side is constant and equal to (Tr,,,0" 0), so that:
fs2
vx dx - 7r,,,0 .
(,)
The same arguments apply to the other invariants. For example, since Ftco0 converges weakly towards fi(x) -- f z dvx(z.), we have, for the energy, ,T(Ftco0) -+ ,~(~), which is the energy of the Young measure v, and thus: = (~) -- ,.,=(coo) We shall denote by (**) the set of constraints (associated to the constants of the motion) other than (,), that fi has to satisfy: = {energy constraint, angular momentum constraint (eventually)].
(**)
Thus we see that the constants of the motion bring the constraints ( , ) , (**) on the possible long-time limits. Since we do not know anything (in the general case) on the long time behavior of the solutions of Euler equation, we will consider Young measures merely as a convenient framework in which we can perform the thermodynamic limit of a family of invariant measures. In the following, we will call the Young measures satisfying the constraints (,), (**) the mixed macrostates, in contrast to the small scale oscillating vorticity functions called microstates.
18
R. Robert
The random Young measure ~fh" Let s be a bounded open subset of R d, the space Fh (s is composed of the functions of the form y~j f J / 3 ( ~ - j ) which are compactly supported in s (see Appendix A). The space Lh (Y2) = Zrh (Fh (s is composed of functions of the
form Z j f/{ X(~ - J) which vanish in a neighborhood of the boundary 0s (whose width 9
X
approaches zero with h). Let us write a function of
(~j~jhdf/, and # h -
89
Lh(g2)" f h - Y~j~Jh f / , x ( - f i - J). We denote d f h f fZdx)dfh, the probability measure on L h ( ~ ) , 9
X
where the scaling factor 1/h a is introduced in order to give a finite value to the mean
fLh(S2)(f f2 d x ) d # h ( f h ) , in the limit h ~ 0.We will write/Zh -- (~)j~Jh dzr*(fhj ) ' where dTr,(y) -- _1__1 vq- e-y2 dy. We will consider now fh as a random function with probability distribution #h- Thus ~i/i, is a random Young measure on s x I~. It follows from Theorem 2.1.6 that the family (depending on h) of the random Young measures ~i/), has the large deviation property with constants 1/ h a and rate function I~ (v), where we denote 7r -- dx | Jr, A straightforward consequence of this large deviation property is that the random Young measures ~i/), which in addition satisfy the constraints (,), (**) are exponentially concentrated about the set g'* of the solutions of the variational problem l~(v*)--inf{l~(~):
~ c ~ ' l,
where g" is the closed subset of the Young measures on .(2 x R satisfying the constraints (,),
(**). Note that this variational problem has at least one solution since ,5' is nonempty and closed and l~r(v) is a lower semicontinuous and inf-compact functional on M. Now, let us denote 7r0 = ~ 1 zrr 0 and 7r f - dx | 7r0. For all v satisfying (,), one can easily get the relationship: l~(v) = l:r,(v) + If2l/zr, (n'0). Thus if G, (Tr0) < oo, minimizing /zr or Izr' on g' gives the same equilibrium set g'*. In fact, the use of the functional /~r, is more natural since it is associated to the invariant distribution 7r,~0. To justify the use of 17r' in the degenerate case/~r, (Tr0) = e~, one can, for instance, modify the definition of the measures Uh, and consider Uh -- (~.i~Sh dTrh (.[)[ ), where dTrh(y) -- 7l exp(--Qh(v))dv, and the polynomial function Qh(Y) is such that zrh converges towards zr0 in the narrow topology when h ~ 0. Of course, we have
/th--~exp
' ( -'- ~f-
Q /, (.fh) dx)
dr),.
It is not difficult to see that the proof of Theorem 2.1.6 [41 ] works for these measures, and it follows that 6./), has the large deviation property with constants l / h a and rate function l~r' (v). Notice that - I j r , ( v ) is the entropy, that is, the functional which measures the disorder created in the fluid by the turbulent mixing.
Statistical hydrodynamics (Onsager revisited)
19
REMARKS.
9 For Euler equation we have d = 2. We shall consider also the case d = 6 for V l a s o v Poisson system. 9 In order to get probability measures, we replace d fh by phdfh with Ph = lZ e x p ( - - ~ f 2 dx) despite the fact that this functional is (eventually) not conserved by the flow. Indeed, we consider as an authorized trick to multiply the measures by any functional which is conserved by the flow of the infinite-dimensional dynamical system.
f
2.1.6. Computation of the equilibrium states, the equation of Gibbs states. Once we have identified the relevant entropy functional, the determination of the equilibrium states come down to the solution of a variational problem: i.e., find the minimum value of Izr'(v) under the constraints given by the constants of the motion of the system. After that it remains to discuss at a physical level the relevance of these states. From now on (for obvious typographical reasons) we shall denote co instead of coo, rr instead of 7r', and K~r(v) = - G ' ( v ) the Kullback entropy. We have now to solve the variational problem: Find the macrostates v* satisfying
where co is any initial vorticity in L~(S-2). We have seen in a previous remark that such v* always exists. We begin with the simpler case where co takes only a finite number of distinct values at . . . . . a , (co takes the value ai on the set S-2i). Then, we have 7r~o = IS2113,~ + . . . + IS2,, 13,,,. It is clear that any mixed macrostate is of the form vx = el (x)6, t + - - . + e,, (x)6,,,, with the constraints:
Fi (el . . . . . e,, ) -- fs2 ei (x) dx -- IS'2il,
i=
l,...,n.
The most mixed state is such that ei(x) --IS2il/IS21, for every x" that is, zrx : T-~-Tzr~,,. The probability distribution Vx is obtained by multiplying rrx by a function equal to ei(x)]F2l/lI-2il for a = ai. This function is equal to dv,:/drrx. Therefore, the Kullback entropy writes as: i
Kjr(v) -- -
~
ei(x)
i
Z
Log
i--~/]ei(x)
t
ei (x) Log ei (x) dx - Z i
dx
IS21).
1s I Log ~ i
i
R. Robert
20
Since the second term is a constant, this entropy is indeed equivalent to the classical Boltzmann mixing entropy"
S(e)-- - f ~ Z
ei (x) Log ei (x) dx.
As e = (e~ . . . . . en) must satisfy the supplementary constraint Z i L 1 ei(x) -- 1, only n -- 1 independent constraints Fl . . . . . Fn- l remain. Let e *I , . . . . e n* be a solution of our variational problem. Then, by the rule of L a g r a n g e multipliers, there are constants c~ -- (c~l . . . . . Otn-l) and fl such that the first variations of the functionals satisfy: n-I
i--1 for all variations 6el such that Y'~i ~ei -- O. Straightforward computations give:
6 s - - - fs2 Z ( l i
+ Loge*)3eidx,
6E -- -- fs2 ~* (~i
ai6ei) dx'
where ~* is the stream function associated with the vorticity ~--~iaiei"*(x),
Fi -- is-2 ~ei dx. Then, we easily get:
e*(x) --
e-Oti -flai Vz*(x ) (l)
z(~*(x))
where the partition function Z is given by
Z(~) = ~ e -~;-~'';v' i=1
and we take C~n -- 0, by convention. Thus ~ * satisfies the equation of Gibbs states:
(G.S.E.)
- - A ~ * = ~--~iaiei*(x)= ~p* - 0 on 0S2.
d Log Z(~p*), fll Or
}
So, we see that if e i* is a solution of the variational problem, there are constants c~, fl such that e* is given by (1) and 7t* by (G.S.E.).
Statistical hydrodynamics (Onsager revisited)
21
Conversely, for any given set of parameters a,/4 we can consider a solution gr c~'/~ of the nonlinear equation (G.S.E.) (as the right-hand side of the equation is of the form .f(O*), with f continuous and bounded, we know, using Schauder's fixed point theorem, that a solution, in general not unique, always exists) and the associated Gibbs state e ~ given by (1). Of course, e ~ is a critical point of the functional J ( e ) = ,n- 1 n S(e) - - ,fl_ ,v ( e ) -- rz-,i=l oti Fi (e) , on the linear manifold E i = l e i (x ) -- I. Furthermore, we prove the following result: P R O P O S I T I O N 2.1.8. /ffl > --Xi/(}--~ia2), where )vl > 0 is the first eigenvalue ofithe ope r a t o r - - A (associated to the Dirichlet b o u n d a r y value condition), then e ~'fi is the unique m a x i m u m o f the.functional J (e) on the set defined by ei (x) >~ O, i = 1. . . . . n, ~ i ei (x) = 1.
PROOF. We shall prove that the functional J (e) is strictly concave on the set defined by 0 < e l ( x ) < 1, i = 1. . . . . n. For that, let us compute the second variation 32j for any
variation 3ei. Straightforward computations give: 1 fs2~-~(fei)2 f'~ S --
2 i=!
f 2 , T , - l fs 2 fig, 6 w d x 2
ei
where we denote f w = E i aifei and fV,~ is the stream function associated to fw. As f2 Fi = 0, it becomes: 32j=
lf2~(fei)2 2 ei i=1
dx - tiff-2] ; 5vl 3wdx.
Let us consider first the case fl ~ (). Then we have, by Green's fl)rmula:
fs2
3 5 3wdx -- ] ~ (V3~)2 dx >~ O,
from where:
l/2~(3ei)2 32 Y <~ - - 2 ei
dx < O,
i=1
thus J is strictly concave. We consider now the case/4 < 0. Using the well-known inequality
~
6 w d x <~ -s
(6w)- dx
o
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Statistk'al h vdrodvmtm i~'s (Onsager revisited)
25
2.2. Extension of the theory to other ,2vstems, the Vlasov-Poisson equation 2.2.1. A c/ass ~?fdynamical systems. A large class of evolution equations, coming from the modeling of various physical phenomena displaying complex turbulent behavior, can be described as the convection of a scalar density by an incompressible velocity field. More precisely, they are of the form:
{
ql + div(qu) - 0, u - - L ( q ), div(u)--0,
where q(t, x) is some scalar density function defined on R • S2 (S2 is a bounded connected smooth domain of R~/), u(t, x) is an incompressible velocity field taking its values in R ~/, which can be recovered from q by solving a RD.E. system. Thus L denotes a (not necessarily linear) integro-differential operator. Let us give some well-known examples of such systems. (1) The simplest example of (1) is the linear transport equatioll, where u(x) is a given incompressible velocity field on ,<2. (2) The quasi-geostn>phic ntodel of geophysical fluid dynamics [40]:
{
q~ + div(qu) - (), u -- curl q,', -At/,, + ~.q~ - q + .l(x),
q,' = () on i)
where q is the so-called potential vorticity, c is a nonnegative constant, and ./ a given function in L"~ (S-2). Obviously, Euler equation is obtained by taking c and ./ equal to (). (3) Collisionless kinetic equations such as the Vlaso~,-Poisson eqttatiott of stellar dynamics can also be written in the form (1). Indeed, let f(t, x, v) be the scalar density function giving the distribution of the stars in a galaxy (in the phase space (x = position, v = velocity)). If collisions (close encounters) are neglected, the density is purely advected by the flow in phase space, so that it satisfies the following Vlasov-Poisson equation:
(V)
af + divx(v 1) + d i v v ( E f ) -- 0,
where E(t x) is the gravitational field, E ,
-Vxq> @(t x) - - G .[' P(t'x' I"dx' is the grav,
,
']X_X
~9
itational potential, and p(t, x) = f f ( t , x, v)dv is the spatial density of stars. Equation (V) conserves the following functionals: The total energy (kinetic + potential): -
,~(.f)--
I/Iv2 ~
(.f(x,v))dxdv+~ If
p(Pdx;
R. Robert
26 -
The integrals
fr -
The linear momentum
ff -
o ( f ) dx dv, for any continuous function 0 such that 0(0) -- 0;
v f dx dv"
The angular momentum
ff
x A v f dx dv.
Let us d e n o t e U may be written:
(~,';)" U is a vector field on R (', which satisfies d i v e , ( U ) -
.1} + div(,(./U)
O, a n d ( V )
= (),
so that Equation ( V ) in of the form (I). The lirst step in our program is to deline a tlow associated to (!) on the phase space L ' (,(2). Untk)rtunately, to our knowledge, there in no general existence-uniqueness result for the Cauchy pn~blem lk~r systems like (l). Examples I and 2 are well known, bt!! c()nccrning Vlasov-R~isson equation, things are more intricate. One can lind existence results for regular initial data in [6 ], while the existence of weak solutions in investigated in [3,2(),28,37]. A nice existence theorem of weak solutions corresponding !c~ a~ initial density ./i~ in I,~(I~ ~') ~ 1,*(1I~'), satisfying ./j'~Iv2 ./ijdxdv < cxs, is given by Horst and Hunze[28]. An usual, uniqueness in more tricky, and we need some additional assumption on the initial density. Let us mention two recent results. - with a Lipschitz regularity condition on ./i) (Lions and Perthame [37]), - with ./i) in L ~ and compactly supported (Robert [5()]). Finally, for our concena, if ./i) in given in I, ~ with compact support, the situation in completely analogous to that given by Yudovich's theorem for Euler equation: We have a unique weak solution .f(l, x, v) in ('([(), ec[: L/'(R(')) for all / 7, I ~< p < cyo, which delines a ttow FI on L'V (R(~). Using the moment estimates of [37], we easily prove that this weak solution remains compactly supported tk)r all time. Here we have also a stability property" I1" ./i~ in a bounded sequence in L < ( I U ' ) , all the functions having their support included in a same compact net which converges in the strong L2 topology towards ./i), then F~./i~ uniformly converges L2-strongly towards Ft./i) on any bounded time interwd. 2.2.2. Fillite-dimensionalapln'oximatiotls o./'Vlasov-Poisson equation. We will consider weak solutions of (V), on a fixed time interval 10, T], corresponding to ./i) in L~(R(~),
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R. Robert
34
The M.E.RP. then gives the following variational problem (V.P. 1):
(J! . . . . . J,,) - max } (JI . . . . . J,,), under the constraints: (C l) ~ i Ji (x) = O, for all x,
(C2) ~" (Jj . . . . . J,,) = 0,
j2 (C3) _ ~ - - ' ( x ) ~ < C ( x )
for a l l x
It can be shown [53] that this problem always has a solution Jj . . . . . J,, satisfying (C3) with equality; moreover, there exist a parameter/4 and a measurable function A(x) such that:
(A)
Ji - - A ( x ) [ V p i
- fi((7)- ai)piV~/I].
This gives a general fl~rm fl~r the currents JI J,,. The variable (~> O) diffusion coefficient A(x) is not known; but once it is given, the parameter fi, which is the Lagrange .
.
.
.
.
multiplier of ,E,, is determined by the conservation of the energy"
/
V (,, 9,I,,, dx - (),
which gives
(B)
/4---
V~/J. VrT)A(x) dx
/L
(Vt/i)2((,)- 2 - ( 7 ) 2 ) A ( x ) d x ,
where o) 2 - ~--~-i ct'[ Pi. Next we need to determine the diffusion coefficient A (x). For this, let us consider the particular case fl = (}. in this case, the energy constraint is not active and Ji is merely an ordinary diffusion current. Thus, we can compute A (x) by using the analogy with the convection-diffusion of a passive scalar. Let us introduce the notations: w=&+&,
u=fi+fi,
& is the fine-scale fluctuation of the vorticity and fi the corresponding fluctuation of velocity. Let us suppose that some scalar density p(t, x) is convected by the microscopic flow u. Then the mean value fi(t, x) of p(t, y) on some ball B(x, r) is convected by the mean field fi and undergoes a diffusion process created by the fluctuation ft. /~(t, x) will satisfy an equation of convection-diffusion type: tst + v . ( ~ s f i + J ) = 0 ,
II
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~ ~
O"
i
::3-
0
~.
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Z=
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II
t~
Ci.
-,.
36
R. Robert
The link between ( R E n ) and the equilibrium theory is given by the following convergence result. PROPOSITION 2.3.1. Let us suppose that the solution p i ( t , x ) , i - - 1. . . . . n, of ( R E n ) converges (in a strong e n o u g h sense), when t tends to infinity, towards a stationary state Pi*(x). Let us a s s u m e also that there is some open c o n n e c t e d s u b d o m a i n A o f F 2 satisfying A C F2 and: (*) Pi* (x) > 0 and A (p*)(x) > O, f o r all x in A. (**) U* 9n -- 0 and J* 9 n - 0 on OA. (n is the unit vector n o r m a l to the b o u n d a r y OA, u* is the velocity field associated to Fo, -- ~ aip*, a n d J~ is the current a s s o c i a t e d to p* by ( A ) . ) Then Pi* (x) is a Gibbs s t a t e on A" that is, there are p a r a m e t e r s ot I , . . . , or, (otn -- O) a n d fl* such that:
Pi
*
(X)
--
exp(-cF - fi*ai~*(x)) Z( ~* (x))
moreover, fl* - - l i m ; _ ~
,
'
for all x in A"
fl(t), where fl(t) is given at each time by (B).
PR()OF. When t --+ cx~, we have: p i ( t , x ) -+ p*(x), &(t,x) --~ &,(x), ~//(t, x) --* r and fl(t) --+ fi* (given by (B) where we replace p i ( t , x ) by p*(x)). We have also Ji(t) -- A ( p ( t ) ) [ V p i ( t ) - fl(t)(rb(t) - d i )pi(t)V~/J] ~ Ji*, and since p i ( t , x) satisfy (REtt), the t'unctions /if (x) will satisfy the set of stationary equations" V . (/,*u* + J * ) - ( ) ,
i-I
. . . . . ,,.
Now let us calculate fA E
Ln I , * V . (/,*u* + Ji*)dx i
-L
E
Ln p* V p * . u* dx + f,a E i
Ln p* V. J* dx. i
Integrating by parts, we see that the first term is zero (due to V 9u* -- ()) while the second gives"
v P7
9Ji* d x -
-
-
fA~ fi*
9 P---~i I[Vz,
fA
* - /~*( , - b , --
ai
)V~*-J*
a i ) l i,* VO*]
9Ji9 dx
dx,
I
but the last term vanishes since y~ J , - 0 and J~a V~p*. J,.* dx - 0 (this last equality comes from .['A ~ * V . (/flu* + J * ) d x - O, by integration by parts).
Statistical hydrodynamics (Onsager revisited)
37
We finally get
fA Z
i
--Z-Z[ 1 Vp* -- ~*(w* --ai)Pi, V ~ * ] A ( p * ) d x - - O, Pi
from where V Ln Pi* -/4*(&,
-
Cli)V~* - - 0
on A,
for i -- 1. . . . . n.
Subtracting equation n from equation i, we deduce that L n ~P; + # ,
(ai --an)~ 9
has a
constant value -or; on A. Now, using the relationship ~ p*(x) -- 1, we deduce that p* satisfy the Gibbs state relationships on A. D The relaxation equations (REn) have been used for large eddy simulations [53]" they where found very efficient.
3. Out-of-equilibrium problems" Weak solutions, shocks, and energy dissipation 3.1. Statisti~'al solutions of ID invis~'id Burgers equation 3.1.1. Definitioll qf statisti~'al s<>httiolls
dt
=a(u),
u(O)--u(),
Let us begin with the
(I)
where a" R" -+ IR" is a smooth mapping, and we assume that for any given initial state (I) has a global smooth solution defined for all time"
u(t)
-
-
%u().
The family Ot is a group of smooth diffeomorphisms of Rn. Then to any given initial Borel probability measure/~() on R" we can associate the family
I~I -- ~t (~()).
(II)
Thus, in this case we clearly know what we mean by statistical solution" a family of Borel probability measures satisfying (II). Let us now introduce the characteristic function
38
R. Robert
Making appropriate integrability assumptions, we calculate the derivative 3,fi, (v) -- i
f
a ( u ) . r e ' " " d/tl(u),
for all v in IR".
(III)
We can easily check here that, under some integrability assumptions, (III) is equivalent
to (II). Let us now consider the more general case where (I) is an ew)lution equation, u taking its values in some functional space. Let us assume that the Cauchy problem for (I) can be solved by means of a continuous semigroup St acting on the space E (with dual E'). The same calculation as above for/it = St (P0) formally yields:
f<.,,,,.
,,, ,,
dp, (u),
for all v in E'.
(III')
We will consider the fl)rmula (III') to give a definition of statistical solutions. More precisely, since we will always consider for E a dense subspace of the space of distributions 79'(R"), we will have CI~ C E'. DI':I:INITI()N. We shall say that t*~ is a statistical solution of(1) if fl~r all ~5 in (7r fit(z~) is a C I function o f l, the right-hand side o f ( l i t ' ) is properly detined and the equality hc)lds. REMARK. At this level of generality, the question of the uniqueness ot" the solutions of (i11') is open. Indeed, even if we suppose the existence of a semigroup $I, we certainly need strong assumptions to prove that a solution of (i11') satisties t*t = & (I*{~).
3.1.2. k)'om ho,u~etzeotts to illtrilzsi~' statisti~'ai s o h t t i o l t s q / B l t r g e r s eqttatiolz.
The one-
dimensional Burgers equation
(B)
iJ t II -~- iJ.v
('
-~ ll
-- ()
!
is of the form (I) with ti(ll) -- --iJ.v(-5l!2). Equation (B) has been extensively studied [58]. We know that starting with a smooth initial data tt(). a unique smooth solution exists for a short time but discontinuities may occur at a linite time. We can also prove that discontinuous weak solutions exist for all time but then uniqueness is lost. To ensure the uniqueness of such weak solutions one has to introduce a supplementary condition (Lax entropy condition which amounts to consider only discontinuities with negative jumps). Finally, Kruzkov's theorem gives a contraction semigroup in the space L I lk~r the weak solution. After integration by parts (III') gives for the general statistical solutions
~}t/~,(V)
-- i
-~u-
ei
") d p , ( u ) ,
for all v in C ~
Statistical tlydrodynanlics (Onsager revisited)
39
where #~ is a family of Borel probability measures on 7)'(IR). For any real number h we denote rh :7)' --+ 79' the translation operator r h ( f ) ( x ) = f ( x -- h). We shall say that/z~ is an h o m o g e n e o u s statistical solution if, in addition, we have for all t: Z'h(ftr =/zr
for all h.
A rather straightforward but important remark is that if #t is a h o m o g e n e o u s statistical solution of (B), it is also a statistical solution of
('
)
ibu + 'a,. -~(u - c) 2 - 0 ,
for all c.
This remark brings us to the following useful proposition: PROPOSITION 3. I. 1. Let t*~ be a homogeneous statistical solution o [ ( B ) sati.sfving (i) j' [Ittl[21.2(,I ) dl~t(u) < oo,.fi)r all t and all compact intervals J" ,
(ii) .l'(u u,
t)2
9
d/zt (u) <~ c..l" w- dx, for all w in C ~ . Then, for any.l"un,'tio,~ p in C~7~ with integral 1, and p , , ( x ) -
a,fi,(,,) ilimf(l(,, (,,p,,))2) --
,,~
F_
--
, v'
,
e
i<,, ' ,,> d t , e l
,Lp( ~,), we have:
(ll) ,
.for all v i~l Cir.
PR()()F. We have -~
--
,
-~lt - , v '
,
e i
,,
)
(.)
e i(,, ",,) d t ~ 1 ( l l )
-
/
(it , p , , ) ( l t
,
, v )e '(''''')
dlt,(lt)
,
and we now prove that
f ( , , , p,,){u, v'}e i<'''''} dt,, (1,)
~. (). 11---+4
For this, let us consider the function ./(11) = f ( r h u , p,,)e i (r/,,,.,,) dt*1 (u). Since l,tr is honlogeneous, ./ is independent of/1, taking the derivative at/1 -- 0 yields"
f(,,, p,;)e'<"."> I,,)+if <,,.,,,,>(,,. ,,')e'<"."> (,,/-0. Thus, we only need to prove that > 0, n--* + r
-2-
s
oo
""
~
,r
~
=8
~.
II
"~
r-, E
~ ..
~
~"
~ o__
I~
JF
",-:
~
r,
,r
"
<
,~.
,_,. ,--.
,',:
,._~.
_m
,..,.
~-.
|l
+
.....
E.
=
+
r, :N , ~.
_._.
_.
'--
.--.
::7.
.D. "'J = =~
r~ ~
=
~
~-.
,...
.-.
Z
N
rI.~
""
-_.
=
E
e,-
=7"
..
r,
--"
~.
i
II
"~"
__
-....
~
-H
- =
~
~
,.<
"2> .--..
'3--
":
_
=
,'m ----.. , . <
--.
_
:z
=
=
~
~
S
=
7
~
=-
~
r~
~
:.
="~
~
F,'~ ::i" ~
..~
a~
~" ~
~
~
-"
=-
~
=
~.~
~
_~_'<~'~.
~
~
I
~ -
o
O
=
~
9
9
-"
~"
--. ~"
E. =
~
-" ~., --h, ~ __. 0
= o o _ P
~
:=r"
=-'" [~-?=
~
-
=
=
"~ ~
~
,%
/A
.~
v~
/A
:.
Io
o.
0
..~
=
=7'
Statistical h win)dvnamir
( Onsager revisited)
41
where we use the notation (u | u : Vv) = (uj uk, OJvk ). The definition of intrinsic statistical solutions then straightforwardly follows: 0 , f i 1 ( v ) - i ,,~or
(u, ,o,, )) | (u - (u, ,o,,))'VP(v)}e i('''v} d/z1(u),
f((u-
for all v in (C0~C~)3. We will now exhibit a class of intrinsic statistical solutions of Burgers equation. To do that, we need some material from the theory of Levy processes. 3.1.3. A rough presentation of LdD, processes. For a clear and comprehensive study of Levy processes we refer to [10,27]. We only want to give here an alternate presentation from the general point of view of probability measures on functional spaces. We define directly a homogeneous Levy process with finite variance on the line as a Borel probability measure t* on the quotient space 7)'(R)/constants, with characteristic functional (defined on C0{ ~ ) of the form
a,,,, where (i) (ii) (iii)
(f r
,) a,),
I ~,(.v) -- ./'x 1,(.v) ds, and the function ~//9R --, C satisfies t/i in C 2 and ~/i(())= (), (/(-u,) -- (J(u,), ~/,, is conditionally of positive type, or equivalently -~/,," is of positive type.
The function ~/, is the Levy exponent of the process. Elementary calculations using Bochner's theorem yield the estimate
]r
- ~/,'(0)~1, I ~< ~ ~,2.
Let us consider the functional
fr
l'or u~ in Ci~"
C is obviously continuous on tile space Cir. It is classical to check (usin,,e the properties (i), (ii) (iii)) that it is of positive type and thus by Minies' theorem it defines a Radon probability measure v on the space "P' 1271. The linear operator / is continuous from C{I~ into Ci~, and its transpose I1"'D' ---+ D'/constants is the primitive operator. Thus exp(.]" ~//(1 v(x) dx)) is the characteristic functional of a Radon probability measure 1' on the space 7)'/constants, image of v by ~1, it is the Levy process with exponent ~/J. Using general arguments, we can get further regularity properties of the Levy process. For any open bounded interval J = ]a, b[, the image of v by the restriction operator to J is actually a cylindrical probability on the space L2 (,I) since C (w) is continuous for the L 2
v
+
--I
<~
~.
=
rlS
I
v
.~.
~-
II
%
<~ ~
~.
z
~c~
v
~"
I
~
~
v
II
I
ID"
II
~_.
m
~.
~
~-
II
-e >.
~"
:::3"
=. .
~.
"~
~
"~ II
v
.
I
v
"L.
v
~
II
'-I
=r
~
~
~"
-
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:3"
-
%.
I
"
--~
,,
,~,
=.~-
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~
m
t';.
'-I
A
I
,
: :,, :~ " ~
--.
,-I
::3"
,~
~.
""
~. ,~ '--I.
~:. ~'.
~
-..,
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: ~
II
9
.~. -~
"-
"--
~,
%
%.
~
.
O --.
9
~-~
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~
;:3
9
o
~,
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~-u
9
O
,_..
9 ~
C3" -'. --"i.
/
~,
~..
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,-I
o
I=3'
O
'~ ,-I 9
::3"...
~
,-1
0"~
,-I
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~.
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c
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~
9
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t~
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~-~- ~
~.
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=~
-
::r'
<~
~
z
c',
~.~'=
"~
,~
c
-..
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II
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~v A
I
I
II
I
I
II
C~
C~
+
II
O~ C~
C~
$
9
~-.
C~
:
+
~.
~
~
.~
f
~.
~
. . . .
9
~ ~
~
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~
'-I
~.
~.~
~
9
~
- ~
~
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,_
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II "~
,-
~
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_,.
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~
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9
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9
= ~
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~
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~"
r
=.-.
-~.. ~
~.~_
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--.
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9
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,---,
9
9 <:P
9
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+
o-
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c~
...,
=.~
9
~
"--
~- =~
-=
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I
~-
o
~ o
9-
+
~
.E
~: leq
~, --
~
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=-
2
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.--
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9
0
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9
>
=
o
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~
_
,.~ ,-.,
o~
9
2
~-
"--
..~
oo ~
,.~0"~
._
..~
;
~
~
~
o
.--0
~ ~
,-.
~-~,o:~--.~
~ o =
~
-~,
~
~
=
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~
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"-
~d~ ~
"~.~
"-
.~
=
=
.~
-
"
~/~ , . ~
~=a~
~
,-'
>
.=._ "
~
..
"B
= =~':-~
=
- -
t~
+
=
=
=
~
H.=
=
.-
A\
~
"~.
Z > ~-'=
~
,~...
~ ~ ~-.==~
~
~
;~
:d ~
-
e-
_>
~
,
~a ~
Z
~
~
=
~
r
~
~a
"~
~
~...~
z
C
~~
~
~
~.
.-
46
R. R o b e r t
Let 99 be any infinitely differentiable function with compact ,support on IR3, even, nonnegatire with integral 1, and ~p~:(~ ) - ~q9( 7 )" Put D~:(u)(x) -- 88J' V~0~(~) 93u(3u) 2 d,,e, where 3u -- u(x + ,e) _ u(x). Then, as e approaches O, the functions D~:(u) (which are in L I (]0, T[ • converge, in the sense qf'distributions on ]0, T[ • 7-, towards a distribution D(u), not depending on qg, and the following local equation q f energy is satisfied
)+div
) - v A ~ l u2 -+- V(VU) 2 -+- D ( u ) -
+p
0.
PROOF. Using Sobolev inclusion of H t in L 6, one easily sees u is in L3(0, T" L 3) and therefore uiuk is in L3/2(0, T; L3/2)" the same for p since, taking the divergence of (1), one gets -- A I ,
--
ik
Oi (l! i It k
),
and if p is the only solution with mean zero, the linear operator t t i t t k continuous on L q for I < q < oo, and so p c L3/2(0, T; L3/2). Now let us mollify Equation (I)" denoting u *: - ~p~,u, f f - ~p~:,l,, ( t l i U ) one has iJ t U ~ -~- iJ i ( II i U ) ~' - - I; A U ~ -JI- V
~
p is strongly
~: - - 99 ~ : * ( u i u )
....
p ~ -- ():
this equation, multiplied scalarly by u plus Equation (I) multiplied by u ~, gives
a, (u. u")+ div((u, u~)u + f l u + l,U ~) + E~-
I,A(u. u ~) + 2 v V u . Vu ~ - (),
where
E~,(t ' x) - i~i(ttitt.i )~ lt.i - grill.jill "~i" Since u e L3((), T: L3), u . u * converges to u 2 and ( u . u~)u + p*:u + p u * converges to (u 2 + 2 p ) u in the sense of distributions on 10, T[ • Moreover, Vu" tends to Vu strongly in L2(](), T[ • thus E , ( t , x ) converges in the sense of distributions towards - a t ( u 2) - d i v ( u ( u 2 + 2 1 , ) ) + v A u 2 - 2 v ( V u ) 2. Another calculation gives f
V~p~ (~) 9 6u(6u) 2 d6
: --i)i(uittjttj But Oi (ujuj)~:
+ 2i~i(ttiuj
Ui - - i)i ( u i ( U j U j ) ~ " ) ,
uj
+ (~i(ujuj )~:tli
2ttitt.ji~iu.!
due to the incompressibility of u.
~rl
=.
~-"
..-..q ~
0"
'.-~
..4
-
"
c
~
~"
P-.-
~
~."
~
.-
~
..]
~
",.:
~'
"~
'-
~
~
o
=:
=
~
,-- =
...
::~
,-.
"
9
~
C)
~'~
v.=
=
~
_.
~
s
~
--'-.
. . . .
--.
~
~._.. ~,--.. -4
~
-
""
._
:/':
9
[~]
~"
,..,.
=
_.
.
~::
""
~
2:1
I-rr'r
..~
-~_~_ _
//~
z'r,r
/A
P__..
~-.~.~
'-<
p_,.
X
r~s
r~s
F,
.PPr
II
"-"
i
-~
./"ry
9 p_.
//X
--..
.rr'r
.rrr
o
3.
<:1
o,....
7:-.
"u
./Y'r
.
/A
"
II
,--,.
,--..
,~,
=~
0" ,....
,-.:
'--"
=
P--.,.
,--..
'-:
,.,.,,.
P--,... ~.
=ob
I~
+
I ,,--
~
I=
~-~
~
Z
r
,.--.
~
~
0
=
<
~
e<
R. Rohert
4g
the lack of smoothness could not lead to local energy creation. In other words, one should have D(u) >~ 0 on ]0, T[ x 7-. It is quite remarkable that this condition is satisfied by every weak solution of NavierStokes obtained as a limit of (a subsequence of) solutions u,: of the regularized equation introduced by Leray [34,35 ]:
div(u~: ) -- 0,
u~: (()) -- 9o~: 9 uo.
For uo given in L2 and s > 0, this equation has a unique C ~ solution u,. The sequence (u,) is bounded in L2(0, T: H I ) A L ~ ( 0 , T: L 2) and a subsequence converges to u, a weak solution of Navier-Stokes, weakly in L2(O, T: H I) and strongly in L3(0, T: L3). But for the regularized equation, one has the local energy balance: (1
,)
+ div
(
,.
1 ,
)
+
,u2+,,(Vu,)2
(),
-
hence ~,(Vu,, )2 converges in the sense of distributions towards
i),
2
-div
((,oe)) u
2
+/'
+l,A2
,~
"
For every inlinitely ditterentiable and nonnegative function ~/J(t,.v), the functional u - - , .j.'l'(Vu)2~/j(t,.v)d.vdt is convex and iower-semicontinuous on the weak space L 2 ((), /': HI), and thus
ff
)!!)~)
(Vu~)
2~/,(1, .,) d., dt >~ ff
( Vu
)2
1/, (t, ., ) d., d/.
which implies lim,_,~)i,(Vu, )" - ~,(Vu) 2 = D(u) ~> (). This fact is well known: see, for example, 136 ]. RI:,MARK. Two natural questions arise at this point: (I) Does there exist a weak solution of Navier-Stokes in the space L2((), 7": H i ) A L'V(0, T: L") with D ( u ) g: ()'? (2) Does the condition D(u) ~> 0 imply uniqueness for weak solutions of NavierStokes? Let us call such weak solutions with D(u) ~> ()"dissipative". in the case of inviscid Burgers equation in one-space dimension, D(u) ~> () coincides with the usual entropy condition of negative jumps, which does imply uniqueness. The fl)llowing proposition shows that the condition D(u) ~> 0 appears naturally fl)r weak solutions of Euler equation.
<
I~
7t.
~
~ --.
,--'.
"<
"~
~
~
~,
"~
~
- -
~
II
"
~
<
0
--"
~ ~
~
~"
P-" ~
"~
-
0
~
--]
~
~
=
-
.-. '
II
9
"="
.
--.
~
'=
.
9- -
_
"~
--"
.
.
I
.
,-,
tJ.~
.'7-" ~,
.~
~
,-
,--
~
=
~
--"
.
E
x
,'--.
m"
+
>-
~..
=_ ~
~
"
9
~
9
Z
"
.
--.
~ -..
. . . .
~
--
~
0
(T
r,
~-
~-~ "~. 9 t.al- ~ E'~ "+. '-~
:~'_
,-~
.-.
0
-=
(T
'-"
m
o
~-
~
~ :m P-'-
~
_.
tu
,.~ ~ -
=---
.-~
~ + 9
~
(T
--.
-
o
qL
, ~ , hal --
~O
T
Z
0 ~
=
~,..~
\V.~
~.
~.
m
"~
;4
~o
9 ~
"
,~
-~ ~'~-
50
R. Robert
integrating in ~ over the ball
O
m
I~1 ~ ~ one gets
'(s
3 im16rr }+()e
(u(x + sse) - u ( x ) ) 2 ( u ( x
+ s~i) - u ( x ) ) , se dZ'(se)).
i=l
Our expression of (u(x + s.se) - u ( x ) ) 2 ( u ( x
s(u) -- lim ~:-+() 8
+ eeq) _ u(.,c)), qedX'(se)
l= I
thus simply gives a local nonrandom form of the above expression of the inertial dissipation.
4. Last comments and acknowledgements We have tried here to give an overview of some topics which we feel inspired by Onsager's views. We l'ocussed as far its possible on issues which yielded rather precise mathematical developments. Doing so we w~luntarily discarded other important issues in turbulence theory such its intcrmittency or the recent important disc~vcry of the inverse energy cascade in 2D turbulence. it in a great pleasure for me to take the ()pportunity to warmly thank those wh(~ have had a direct contribution in the elaboration of the present work" F. Bouchet, P.H. Chavanis, J. Duchon (who greatly helped me to write Section 3. ! ), T. Dtlmont, J. Michel, A. Mikelic, C. Rosier, J. Sommeria. Great thanks also t(7 C. Bardos, B. Castaing, P. Constantin, A.J. Chorin, G. Eyink, M. Farge, U. Frisch, J.L. Lebovitz, A. Majda, R. Moreitu, who kindly contributed t() the discussion and diffusion of these ideas.
Appendix A
Ftitile-elellseitt ai~pro.vi, taliolt. F()r the comfort of the reader, we briefly recall some standard notations and properties 141. Apl~ro.vi, tatiolz o/ tile Sohole~' space H'"(IR a). We denote Q,/ - l - I/2, I/21 '1, Z the characteristic function of Q,/. and fi - X , ' " , X (,z + I terms). For a given parameter h > (), we define a prolongation operator I'/," to any function ./), -- y-~.i ./)ii X (7; .j) (.j beX
longs to Z '/), we associate the function
p,,.i>,- .li[t
(x) T-.J.
__
~
,..
.5
:>
9~
~
.
~
.>-
.~
,..a
.
9
~.
....
.....
...
<
.----- .~.
~'-*>
~
...~
E
to
~g
~>~>~>
.
-:~"~ > ~ . < ~ <
,..,.
.~
~I
-
---
a:
.<
/A
-4
//'x
._.
~
"-
~
r
----
---
-
.~
'.-2
~
7"
,...
P-,...,.
;>
.....
II
v....
,_0
~
"" ~ "
Z..
~ ~-.
,..,.<'6
-
~"
~
~,"
=a
~'~
~
2
r~
"*
~
"
---
/A
a:
9'
/A
"~
o
,r
~
:a-"
~
"a
"~~
9
.<
o~--
~
~
~-
~,,,
,ca
~
~
{'D
=
II
--" 9
9
m
,r
-"
~
q" ~
I
"
A
~
~
II
", =
~~
o
=:
'-<
52
R. Robert
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