Compact attractors for the Navier-Stokes equations of one-dimensional, compressible flow

5. R. Acad. Sci. Paris, t. 328, SCrie I, p. 239-244, Equations aux d&i&es partielles/fartial Differential (Probkmes mathCmatiques de la m&zanique/Mathematical

Compact attractors of one-dimensional, David

HOFF

a, Mohammed Indiana

” Department of Mathematics, E-mail: [email protected]

Texas

(Requ

le 4 septemhre

Abstract.

1998,

in Mechanics)

equations

b

University A&M

University,

le l”?

dkcemhre

Bloomington, College

IN

47405,

Station,

TX

US.4 77843,

USA

1998)

We prove the existence of a compact attractor for the Navier-Stokes equations of compressible fluid flow in one space dimension. We also show that the large-time behavior of a given solution is entirely determined by its values for all time at a finite number of points, given in terms of a certain dimensionless quantity associated with a canonical scaling of the system. Our results are based on a well-posedness theory for these equations which goes beyond previously known results. In particular, we establish the global existence and regularity of solutions with large external forces and large, nonsmooth initial data, with regularity estimates independent of time. 0 AcadCmie des Sciences/Elsevier, Paris Attructeurs d ‘un jluide

R&urn&

accept6

Problems

for the Navier-Stokes compressible flow ZIANE

a Department of Mathematics, E-mail: [email protected]

1999 Equations

compacts pour les bquations compressible monodimensionel

de Navier-Stokes

Dans cette Note nous &tab&sons l’existence d’un attracteur compact pour les iquations de Navier-Stokes gouvernant unJuide compressible en une dimension de l’espace. Nous montrons aussi que le comportement ir l’infini des solutions est entiPrement dPterminP par la connaissance de la solution en tout temps sur un ensemble$ni de points, dont le nombre est major& par une constante associPe ir une renormalisation canonique du systkme. Nos r&.dtats sent basis sur une tht?orie d’existence qui va au de& des r&ultats connus. En particulier, nous montrons 1‘existence globale de solutions borne’es pour de grandes forces et don&es initiales peu r&ditre.s. 0 AcadCmie des ScienceslElsevier,

Paris

Version frangaise abr6g6e Dans cette Note nous Ctablissons l’existence d’un attracteur compact pour les equations de NavierStokes gouvemant un fluide compressible en une dimension de l’espace. Nous montrons aussi que le comportement B l’infini des solutions est entikrement determine par la connaissance de la solution en tout temps sur un ensemble fini de points, dont le nombre est majort par une constante associCe

Note prCsentCe par Pierre-Louis

LIONS.

0764-4442/99/032802390 AcadCmiedes ScienceslElsevier, Paris

239

D. Hoff, M. Ziane

a une renormalisation canonique du systeme (voir 131,[5]). Nos resultats sont basessur une theorie d’existence qui va au dela des resultats connus. En particulier, nous montrons l’existence et la regularit& globale de solutions pour de grandes forces et donnees initiales peu regulieres. De plus, nous Ctablissonsdes estimations uniformes qui sont suffisantes pour obtenir la dissipativitt n6cessaire pour l’analyse asymptotique des solutions quand le temps tend vers l’infini. Du fait de la persistance des discontinuitts dans la densite pour tout t > 0: le semi-processus associe au systeme n’est pas compact. Cette difficult6 est resolue par I’exploitation d’un certain effet hyperbolique qui regularise la densite en temps infini, c’est-a-dire que le semi-processusest asymptotiquement compact. Dans le but de simplifier la presentation, nous decrivons seulement nos resultats dans le cas simple ou la densite initiale est dans l’espace de Sobolev Hi. Le cas general ou la densite est dans l’espace de fonctions a variations born&es, est trait& dans 181.

1. Introduction We prove the existence of a compact attractor for the Navier-Stokes equations of compressible fluid flow in one space dimension. We also show that the large-time behavior of a given solution is entirely determined by its values for all time at a finite number of points, given in terms of a canonical scaling of the system. Our results are based on a well-posedness theory for these equations which goes beyond previously known results. In particular, we establish the global existence and regularity of solutions with large external forces and large, nonsmooth initial data. We obtain bounds which are time-independent, sufficient to establish the dissipativenessrequired for an analysis of the large-time behavior. We note here that initial discontinuities in the density and its derivatives persist for all time, so that the solution operator for this system is not compact in forward time. We overcome this difficulty in the attractor theory by exploiting a certain hyperbolic effect, which gives smoothing of the density in injnite time, resulting in a kind of “asymptotic compactness.” These are the first results proving the existence of an attractor for the Navier-Stokes equations of compressible flow. For the case of incompressible flow, the density enters in a trivial way, the solution operator is compact, and the underlying well-posednesstheory is far more developed. In particular, the existence of a compact attractor is known in two, but not three spacedimensions(seefor example [4]). The Navier-Stokes equations for compressible flow on a finite interval express the laws of conservation of mass and balance of momentum as follows:

Pt + (W)E = 0, (PU)t + h2).1: + (P(P)>z = u.cz+ pf(n:, t)! p(.,O) = (20, u(.;O) = uo, 71= 0: z = 0, 1,

1 pc(z)d.r = 1. ./ 0

(1.1) (1.2) (1.3) (1.4)

Here t 2 0 is time, z E (0,l) is the spatial coordinate, and p(z, t), U(Z, t) and P(p) represent, respectively, the fluid density, velocity and pressure,and f = f(~? t) is a given force density. We have scaled out all time and length parameters, so that the quantities appearing above are dimensionless, and the spatial interval and the total mass are normalized. We take P(p) = A2p, where A is the result of these scalings. Specifically, A is the sound speed times the total mass divided by the viscosity coefficient.

240

Attractors

for compressible

Navier-Stokes

equations

In the present Note we describe our results only for the somewhat special case that the initial density is in H1(O, l), so as to avoid certain technical details. In [g] we consider the more general and more difficult case that the initial density is in BV, so that propogation of singularities plays a role, and a richer dynamics occurs. These extensions are described in general terms below, following the statements of Theorems 2.1 and 3.2. We assume for the present that the initial data (pa, ~a) satisfies uo E L2,

PO

E H1, with pa > 0, pi1 E L”,

(1.5)

and that the force f satisfies f E W1@(R+;L2);

i.e., Ilfll~--

= II.~IIL-(R+;Lz)

+ Ilftll~=(~+;~~)

< 00.

More generally, we let C be a bounded, translation invariant subset of W1;“(R+; for f E Z llfllc = llfll~-. 2. Uniform

estimates

and global existence of bounded

(1.6)

L2) and define,

solutions

In this section we describe our results concerning the existence and regularity of solutions of (l.l)-(1.4). We let Co denote a constant depending on IuaI~z~ (I~cl(n~, llp;1j\L55, and Ilfllc, and we define explicitly constants C, and Kf in terms of f and A. Also, for a given pair (p; u), we define the functionals:

where $ is the potential-energy density $(p) = p log p - p + 1. We denote by I . I the usual norm in L2((0, l)), and identify other norms explicitly as they arise. We then have the following existence and regularity results for solutions of (l.l)-(1.4). THEOREM 2.1. - Assume rhat the initial data (~0, ~0) satisfiesthe conditions in (1 S) and that f E C. Then there exists a unique global weak solution of (l.l)-(1.4) satisjjGng: P E C([O, m); H1),

14$)12 ; lt

+ Ilp(-,t>ll;l

u E C([O:cc);L2) + ll~-~(.,t)ll;-

(u,(.> s)l’ds 5 4Cf + %,

;~(~,.:.s))2d*(~+$

nL:,,([0,~);H1),21(.,t) + f”

t

vt>o,

Vt>O.

Iwz(v)12dS

E H2, I Co,

V t > 0,

Qt>O,

(2.1) (2.2) (2.3) (2.4)

Moreover, given R > 0 and M > 0, there exists a time T = T(R, M, A) such that, if II f IIx <_M and ‘d t L T. .WO,WI) I R, then -Q(.,t),u(.,t)) I Kf,

Observe that Theorem 2.1 establishes the existence of an “absorbing ball” for the solution operator. This fact, which is a consequence of the estimates (2.2)-(2.4), plays a crucial role in the construction of the attractor. 241

D. Hoff, M. Ziane

In [8] we prove a more general existence result for the case that p. E BV rather than H1 (slightly stronger hypotheses on the force f must be made). Discontinuities in p and in u, now persist for all time, and the derivation of time-independent, a priori bounds requires more work. Two important mechanisms enter prominently into the analysis. The first of these is the exponential-in-time decay of singularities in p and in U, resulting from the hyperbolicity of the underlying inviscid Euler equations. The other is a one-sided l/t bound for the so-called “effective viscous flux” ‘zL,~- P(p), reminiscent of the familiar entropy condition for convex, scalar conservation laws. These effects are combined in a fairly complicated, coupled set of a priori bounds for appropriate approximate solutions, and a rather delicate analysis is again required to close these bounds without smallness assumptions. We remark that the system (1. l)-( 1.4) has been studied by a great many authors; see for example [I], [7], [ lo]-[ 121 and the references contained in these papers. The results of Theorem 2. I and [S] go beyond each of these in some essential way. Theorem 2.1 allows us to define a family of semi-processes {U,(t, T)},,,,, fan on the metric - - , space X = {(p? u) E H1 x L2; p > 0, p-l E L”}, as follows: Uf(t,~-) : X -+ X,

where b(t), u(t)) is the solution of (l.l)-( 1.4), with initial data (p, U) given at time r. We can also define the functionals E(t) = E(ZAf(t, ~)(p, 2~)): B(t) = B(Llf(t,‘r)(p, u)) and *J(t) = .qzqt, r)(p, u)). w e re q uire the following regularity properties of the semi-process Uf(t, T); these are consequences of the estimates (2.1)~(2.4) of Theorem 2.1. THEOREM 2.2. - (i) (Strong continuity) For all jixed 7 > 0, f E C and (p, U) E X, the map t I-+ Uf(t, r)(p, u) is continuousfrom [7! W) into X with respect to the H1 x L2-topology. (ii) (Weak continuity) If fj -+ f strongly in C and (pj : ZL~)-+ (p! u) in the weak topology ofH1 x L2. then UfJ (t, r)(pj, uj) + Uf(t, ~)(p, u) weakly in H1 x L2, V t 2 T 2 0.

(iii) (Uniform absorbing set) There exists a bounded, uniformly absorbing set f3 in X. That is, for any 7 2 0 and any bounded subset B C X, there exists a time T(r, B) 2 T such that 1Af(t,~)B C 13 for t 2 T(r: B) and for all f E C. 3. Asymptotic

compactness

and the existence of a compact

attractor

The existence of a compact attractor for solutions of a particular system of differential equations generally requires that its solution operator be compact. This compactness is typically a consequence of dissipative mechanisms in the system. For the present problem, however, there is no dissipation for the density, and the solution operator is not compact. We can overcome this difficulty by showing instead that the semi-process Uf (t, T) is “asymptotically compact”. This asymptotic compactness results in the present context from the hyperbolicity of the underlying inviscid system (Euler equations) corresponding to (l.l)-(1.4), reflected in the weakly dissipative term on the left side of (2.8). The proof of asymptotic compactness is given in the following theorem. For the more general case that p. E BV, the proof depends upon this hyperbolicity in a much more striking way; see [8] for details. See also [6], [9], and [13] for asymptotic compactness results in other contexts. THEOREM 3.1. - Assume that C is compact. Then the family of semi-processes {Uf (t, r)},,,,, fEc is uniformly asymptotically compact. That is, for (“Oj}jEN

242

= { (POj7 WJj)}je~

bounded in X, {fi},,,

C C,

{tj}j

C R+?

&n tj = 00, J-+-=

Attractors

for compressible

the sequence{h(h), uj(Ci)))iEN = {u.f3(tj, O)u~j}~~~ is precompact

Navier-Stokes

equations

in X with respect to the

topology of H1 x L2. Proof. - We give a brief sketch of the proof. Complete details, including the proof for the case that p E BV, are given in [8]. Without loss of generality, we may assume that {!YQ}~~~ is in the absorbing set of the semi-process, where Ip-’ 1~~ is bounded (independently of the initial data). Thus, A2

there exists a constant X > 0, independent of j such that kjPrllr.X a smooth function cpx such that

l [IVA(P)zlZ - Xpu2 +

5 x.

This allows us to find

u;] dz - X .i’ [ yA2$(p)]

dz.

Note that since C is compact, there exist f, f~ E C and a subsequence j’ such that fj, -+ f in C and fjJ(t - T) -+ fi(t) in C (strongly) for all T E N. Furthermore, using (2.2) and (2.3), we have as j’ + oc,

(Pjf@jf)l Ujr@jr)) = uf3,Ct.jr,O)uOj(

W

+

and (&&),uj~(tj~))

=&&I

- T,O)Uo,, -+ WT

weakly in H1 x H1, for all T E N. Using the uniqueness of solutions and the weak continuity the semi-process, we conclude that: W = iY,,.(T, O)WT>

VTEN

and

dt We write the energy equation as dE(t) formula we obtain

]]Wl( ~1~~2 5 lim,inf IIU~~,,(~~~,O)UO~~IIH~~L~. + XE(t) + K(t)

= J(t).

of

(3.1)

By the variation of constants

T

E(pj,(t>),

ujt(ti))

=E(ls,,(tj~),~j~(tj~))e-XT

+ eCX’

J

ex”J(/7jt(s);

$t(s))ds

0

- epXT

J

T
(3.2)

0

Since Jpj,IL- and ]p;‘]t,are bounded independently of t and j’, and (ipx(~j!)~)~, and (2(jt,)j, are bounded in L2(0, T; L’), we can show that there exists a subsequence still denoted j’ such that cpx(pij) -+ cpx(pT) weakly in L’(O,T;Hl) and UifZ + uTZ weakly in L2(0,T; L2), where @T(s), u;(s)) = zAfT(s: O)Wr. Therefore, T

limsup E(pj,(ti):uj,(ti))

5 Moe-XT + e-XT

j'-+oO

J m(S))ds J eX"~(p~(~), e’“J(m(s)?

w(s))ds

0

_ e-XT

T

,

(3.3)

0

where Ma depends on the size of the absorbing set. Using the energy equation and (3. l), we deduce .+ E(W). Letting T go to from (3.3) that limsupj,,, E(pjr(tS), uj,(t$)) 5 (MO + E(WT))e-“’ co, we conclude that limsupj,,, E(pjr (ti), ZL~,(t$)) 5 E(W). Using the strong convergence of ~~~ in L2, we conclude that

243

D.

Hoff,

M.

Ziane

which together with (3.1) and the strong convergence ;&

The result then follows space H1 x L2.

I11Af3(tj!O)UUj((H1xL”

from the convexity

of uj in L2 implies =

IIWIjH’xLz.

and boundedness

of the absorbing

set in the reflexive cl

The existence of a compact attractor now follows from Theorems 2. I, 2.2, and 3.1, together with the theory developed by Chepyzhov and Vishik [2]. The precise result is as follows: THEOREM 3.2. - Let C be a compact subset of W ‘+(R+; L’). Then there exists a set dc c X which is compact in the topology of H1 x La, and which is uniformly attracting for the semi-process 24, (t, r) associated with the system (1 .l)-( 1.4). That is, ,for all bounded sets B c X,

lim sup dist,y (IAf(t, r)B: d) = 0, t-w fGC

V 7 2 0.

(3.41

We show in [8] that the attractor is characterized by a finite number of determining modes, thereby giving an indication that the asymptotic behavior of solutions of the Navier-Stokes equations of compressible flow is essentially finite-dimensional. We also show in [8] that the attractor is contained in H1 x Hz. This gives an explicit, quantitative description of its compactness, even for the case that X C BV x L2. We recall in this regard the results of [7], which show that, when the initial density is in BV, discontinuities in both p and U, persist for all time, and that (p(., t), u(., t)) E (BV - Hi) x (Hi - H”) for t > 0. The magnitudes of the discontinuities in p and uZ decay exponentially in time, however, so that, speaking loosely, they should disappear after infinite time, and the solution (p, U) should be time-asymptotically smoother than BV x Hi. Our result that the attractor is contained in H1 x H” thus expresses these observations in a precise and rigorous way. It is instructive to contrast the parabolic smoothing, which takes (p, U) from BV x L2 into BV x H1, and which occurs instantly in time, with this “hyperbolic smoothing”, which takes (p, U) from BV x L2 into H1 x H2, but which occurs only in injnite time.

References [1] Amosov A.A., Zlotnick A.A., Global generalized solutions of the equations of the one-dimensional, Soviet Math. Dokl. (1989) l-5. [2] Chepyzhov V.V., Vishik MI., Attractors of non-autonomous dynamical systems and their dimensions, J. Math. Pures Appl. 13 (1994) 279-333. [3] Cockbum B., Jones D.A., Titi ES., Determining degrees of freedom for nonlinear dissipative equations. C. R. Acad. Sci. Parts 321 s&e A (1995) 563-568. [4] Constantin P., Foias C., Temam R., On the dimension of the attractors in two-dimensional turbulence, Physica D 30 (1988) 284-296. [5] Foias C., Temam R., Sur la determination d’un Ccoulement fluide par des observations disc&es, C. R. Acad. Sci. Paris 295 s&e A (1982) 239-241. [6] Ghidaglia J.-M., A note on the strong convergence towards attractors for damped forced KdV equations, J. Differ. Eq. 110 (1994) 356-359. [7] Hoff D., Global existence for lD, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Sot. 303 (1987) 169-181. [8] Hoff D., Ziane M., (in preparation). [9] Ladyzhenskaya O.A., Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991. [IO] Matsumura A., Large-time behavior of one-dimensional motions of compressible viscous gas, in: Recent developments in theoretical fluid mechanics (Paseky, 1992), Vol. 291, Pitman Res. Notes Math. Ser., 1992, pp. 103-128. [I 1] Matsumura A., Yanagi S., Uniform boundedness of the solutions for a one-dimensional isentropic model system of compressible viscous gas, Comm. Math. Phys. 175 (1996) 259-274. [12] Serre D., Solutions faibles globales des equations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris 303 SCrie I (1986) 639-642. [13] Temam R., Infinite-dimensional dynamical Systems in Mechanics and Physics, 2nd Edition, Springer-Verlag, 1997.

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