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Global solution of the pressureless gas equation with viscosity
Physica D 163 (2002) 184–190
Global solution of the pressureless gas equation with viscosity Azzouz Dermoune a,∗ , Boualem Djehiche b a
Laboratoire de Probabilités et Statistique, UFR de Mathématiques, USTL, Bât. M2, 59655 Villeneuve d’Ascq Cédex, France b Department of Mathematics, KTH, 100 44 Stockholm, Sweden Received 9 July 2001; received in revised form 2 January 2002; accepted 7 January 2002 Communicated by U. Frisch
Abstract We construct a global weak solution to a d-dimensional system of zero-pressure gas dynamics modified by introducing a finite artificial viscosity. We use discrete approximations to the continuous gas and make particles move along trajectories of the normalized simple symmetric random walk with deterministic drift. The interaction of these particles is given by a sticky particle dynamics. We show that a subsequence of these approximations converges to a weak solution of the system of zero-pressure gas dynamics in the sense of distributions. This weak solution is interpreted in terms of a random process solution of a nonlinear stochastic differential equation. We get a weak solution of the inviscid system by tending the viscosity to zero. © 2002 Elsevier Science B.V. All rights reserved. PACS: 02.30.J; 02.30; 05.45; 05.40 Keywords: Pressureless gas equations with viscosity; Nonlinear diffusion process; Weak convergence
1. Introduction To explain the formation of large scale structures in the universe, Shandarin and Zeldovich [10] suggested the following toy model of sticky particles: consider a system of particles {xi0 } ⊂ Rd with initial velocities {vi0 } and masses {m0i }. The particles move with constant velocities unless they collide. At collision, the colliding particles stick and form a new massive particle. The laws of conservation of mass and momentum tell us that the probability distribution ρ(dx, t) and the velocity u(x, t) of this new particle are solutions of the following system of equations: ∂ρ + div(uρ) = 0, ∂t (S(0)) = ∂(uρ) + div(u ⊗ uρ) = 0, (∗) ∂t ρ(dx, t) → ρ0 (dx), u(x, t)ρ(dx, t) → u0 (x)ρ0 (dx), weakly as t → 0+ , ∗
Corresponding author. Tel.: +33-3-2043-6515; fax: +33-3-2043-6774. E-mail address: [email protected] (A. Dermoune). 0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 2 ) 0 0 3 5 4 - 8
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called pressureless gas without viscosity. To solve this system of equations when d = 1, E et al. [7] devise a generalized variational principle while Dermoune [3] uses a probabilistic approach by constructing a stochastic process that satisfies the following nonlinear stochastic differential equation (SDE): dXt = E[u0 (X0 )|Xt ] dt,
L(X0 ) = ρ0 (dx).
In d-dimensional space, for any d > 1, the major difficulty of sticky particles is that point particles will generically move without collisions, thus generating a complicated multi-stream flow. In multidimensional case (see, e.g., [9]), the existence problem for zero-pressure gases is believed to be ill-posed. Nevertheless, in this paper, we will modify the Zeldovich model of sticky particles by assuming that the mass and momentum of the new particle have equal diffusion rates. Thus, the probability distribution ρ(dx, t) and the velocity u(x, t) of the new particle are weak solutions of the following system of gas equations with viscosity: ∂ρ ν2 + div(uρ) = ρ, ∂t 2 ν2 (S(ν)) = ∂(uρ) (∗∗ ) + div(u ⊗ uρ) = (uρ), ∂t 2 ρ(dx, t) → ρ0 (dx), u(x, t)ρ(dx, t) → u0 (x)ρ0 (dx), weakly as t → 0+ . In a previous paper (cf. [5]), we constructed a local strong solution of Eq. (∗∗) by solving the following nonlinear SDE: dXt = E[u0 (X0 )|Xt ] dt + ν dBt ,
L(X0 ) = ρ0 (dx),
(1)
where Bt is a standard Brownian motion independent of X0 , up to some time horizon T0 . This was done using a time reversal device of a forward diffusion process, which brings back the problem to the study of a system of quasi-parabolic equations whose solution exists only locally. The objective of the present work is in fact to construct a global weak solution of Eq. (∗∗) given any time horizon T , again by solving Eq. (1). The solution (ρ(dx, t), u(x, t)) of Eq. (∗∗) is then explicitly given as follows: ρ(dx, t) is the probability distribution of the diffusion process Xt at time t and the velocity vector is u(x, t) = E[u0 (X0 )|Xt = x]. In the simple case when X0 takes two values x1 , x2 , our nonlinear diffusion process given by Eq. (1) evolves as follows. Initially, we have two particles located at x1 and x2 , with masses m1 = P (X0 = x1 ), m2 = P (X0 = x2 ), and velocities v1 = u0 (x1 ), v2 = u0 (x2 ). Immediately each particle at xi diffuses with drift u(x, t) = α(x, t)v1 + (1 − α(x, t))v2 , where α(x, t) = P (X0 = x1 |Xt = x) is the proportion of diffusion trajectories starting from x1 and arriving to x at time t. (n) We approach the problem by constructing a system of particles whose positions (Xt , t ∈ [0, T ]) satisfy the following equation: t (n) (n) (n) (n) Xt = X0 + E[u0 (X0 )|Xs(n) ] ds + νBt , t ∈ [0, T ], (2) 0
186
A. Dermoune, B. Djehiche / Physica D 163 (2002) 184–190 (n)
(n)
where X0 is a random variable which has a finite range, and (Bt , t ∈ [0, T ]) is the properly normalized linear interpolation of the simple symmetric random walk on Zd . For each n, ν > 0 fixed we set (n)
(n)
un,ν (x, t) = E[u0 (X0 )|Xt
= x],
and suppose that for all ν > 0, there exists a function uν (x, t) continuous with respect to the variable x, dt a.e. and such that, for each M > 0, lim
sup |un,ν (x, t) − uν (x, t)| → 0,
n→+∞ |x|≤M
t > 0.
(3) (n)
By a standard limit theorem, we will show that the sequence (Xt , t ∈ [0, T ]) converges weakly in D([0, T ], Rd ), the space of functions on [0, T ] with values in Rd which are continuous from the right on [0, T ) and have left hand limits on (0, T ] to a process X(ν) satisfying Eq. (1). We will also show that any limit X of X(ν) as ν → 0 is solution of a differential equation dXt = ϕ(t) dt. If ρ(dx, t) = P (Xt ∈ dx) denotes the probability distribution of Xt and u(x, t) = E[ϕ(t)|Xt = x], then (ρ(dx, t), u(x, t)) is solution of the inviscid gas equations if and only if for every 1 ≤ j ≤ d, E[ϕj (t)u0 (X0 )|Xt ] = E[ϕj (t)|Xt ]E[u0 (X0 )|Xt ] dt ⊗ P
a.e.
(4)
Here ϕj (t) are the components of the vector ϕ(t). In Section 2, we state the main results of the paper. Our system of sticky particles is discussed in Section 3. Finally, in Section 4, we prove the main result, by characterizing the limit process of the system of sticky particles, as solution to Eq. (1).
2. Main results We will denote by ·, · the scalar product in the Euclidean space Rd . By a weak solution of Eq. (∗∗), we mean the following definition. Definition 2.1. A family (ρ(dx, t), u(x, t)ρ(dx, t), t ∈ [0, T ]) of Borel measures is a weak solution of Eq. (∗∗), if for any f ∈ C02 , the space of C 2 -functions with compact support, and any 0 < t1 < t2 : t t (D1) f (x)ρ(dx, t2 ) − f (x)ρ(dx, t1 ) = t12 ∇f (x), u(x, t)ρ(dx, t) dt + (ν 2 /2) t12 f (x)ρ(dx, t) dt; t t (D2) f (x)u(x, t2 )ρ(dx, t2 ) − f (x)u(x, t1 )ρ(dx, t1 ) = (ν 2 /2) t12 f (x)u(x, t)ρ(dx, t) dt + t12 ∇f (x), u(x, t)u(x, t)ρ(dx, t) dt; (D3) ρ(dx, t) → ρ0 (dx), u(x, t)ρ(dx, t) → u0 (x)ρ0 (dx) weakly as t → 0+ . Now we state the main result of the paper. Theorem 2.2. Let X be any solution of dXt = E[u0 (X0 )|Xt ] + ν dBt . We denote by ρ(dx, t, ν) the probability distribution of Xt , and u(x, t, ν) = E[u0 (X0 )|Xt = x]. Then (ρ(dx, t, ν), u(x, t, ν)ρ(dx, t, ν), t ∈ [0, T ]) is a weak solution of Eq. (∗∗).
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Theorem 2.3. Assume that x → u0 (x) is bounded, continuous and ρ0 (dx) a probability distribution on Rd . Under the assumption (3), for all T > 0, there exists a probability measure on C([0, T ], Rd ) under which the coordinate process Xt satisfies dXt = E[u0 (X0 )|Xt ] dt + ν dBt ,
t ∈ [0, T ],
L(X0 ) = ρ0 (dx),
where B is a d-dimensional Brownian motion. Theorem 2.4. Under the assumption (4), there exists a sequence νn → 0 such that the sequence (ρ(dx, t, νn ), u(x, t, νn )ρ(dx, t, νn )) converges weakly to (ρ(dx, t, 0), u(x, t, 0)ρ(dx, t, 0)) solution of zero-pressure gas equations, Eq. (∗), without viscosity.
3. A system of sticky particles 0 } ⊂ R d , with initial velocities Let us first study a system Σ of N particles initially located at {x10 , . . . , xN 0 0 0 u0 (x1 ), . . . , u0 (xN ). Each particle xi has the mass mi . Sticky particles dynamics for the system Σ can be described by the following system of ordinary differential equations:
lim
ε→0+
Xt+ε − Xt = f (Xt ), ε
t > 0,
(5)
where f is the map from RNd to RNd defined by N j =1 mj u0 (xj )1[xj =xi ] fi (X) = , 1 ≤ i ≤ N, N j =1 mj 1[xj =xi ] where f = (f1 , . . . , fN ), X = (x1 , . . . , xN ) are the coordinates, respectively, of f and X. We show easily that for each X0 ∈ RNd the system (5) has a unique solution which starts from X0 . If we denote by t → ϕ(t, X0 ) the latter solution, then the map ϕ : R+ × RNd → RNd satisfies the flow property: (1) ϕ(0, X0 ) = X0 , (2) ϕ(t + s, X0 ) = ϕ(t, ϕ(s, X0 )), s, t ≥ 0. 0 ), then ϕ(t, X ) is the location of the system Σ at time t. The particle, initially located If we take X0 = (x10 , . . . , xN 0 0 at xi , is located at ϕi (t, X0 ) at time t. Now we give a probabilistic interpretation of the flow ϕ. Suppose that N i=1 mi = 1, and consider the proba 0 ). Each random variable X defined from the bility space (Rd , B(Rd ), ρ0 (dx)), where ρ0 (dx) = N m δ(x − x i i=1 i 0 )) ∈ R Nd . Using probability space (Rd , B(Rd ), ρ0 (dx)) to Rd can be seen as a vector X := (X(x10 ), . . . , X(xN 0 0 this identification the vectors X0 = (x1 , . . . , xN ), ϕ(t, X0 ) = (ϕ1 (t, X0 ), . . . , ϕN (t, X0 )) represent, respectively, the random variables X0 : xi0 ∈ Rd → xi0 , and xi0 ∈ Rd → ϕi (t, X0 ). It follows from (5) that the process t → ϕ(t, X0 ) := Xt satisfies the following SDE:
dXt = E[u0 (X0 )|Xt ], dt
(6)
where E[·|Xt ] is the conditional expectation with respect to the random variable Xt . Remark that the flow ϕ(t, X0 ) := ϕ(t, X0 , u0 , ρ0 ) depends only on the velocity u0 and the probability distribution ρ0 (dx) := P (X0 ∈ dx) of the random variable X0 . We will adopt this notation in the sequel.
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Now we come back to the construction of (2). Before doing that we need some notations. Let Ω0 := ({−1, 1}d )N be the set of sequences (en : n ≥ 1) with values in {−1, +1}d . The σ -algebra F0 is spanned by the coordinate (e ∈ Ω0 → πn (e) = en : n ≥ 1). The space (Ω0 , F0 ) is endowed with the probability ⊗N
1 . µ= d 2 d e∈{−1,1}
We consider the product space Ω := Rd × Ω0 , endowed with the σ -algebra F generated by the set B × A for B ∈ B(Rd ) and A ∈ F0 , and the probability measure P := ρ0 (dx) ⊗ µ. Let us construct for each integer n a process satisfying (2) in the probability space (Ω, F, P ). We consider a system of particles with initial locations {xi0 : 1 ≤ i ≤ N } which evolves according to the following dynamics. At time zero the particles are located at (xi0 ), with assigned velocities or drifts (u0 (xi0 )), and masses (mi ). During the interval of time [0, 1/n) this set of particles moves following the sticky particle dynamics given above by the flow ϕ(t, X0 , u0 , P (X0 ∈ dx)). At time t = 1/n each particle ϕi (1/n, X0 , u0 , P (X0 ∈ dx)) jumps to √ ϕi (1/n, X0 , u0 , P (X0 ∈ dx)) + ν/ ne1 , and we get a new system of particles X1/n := ϕ(1/n, X0 , u0 , P (X0 ∈ √ dx)) + ν/ ne1 . At each location x = X1/n there is a massive particle with mass m(x) = P (X1/n = x), and velocity
1 = E[u0 (X0 )|X1/n = x]. u x, n Now again during the interval [1/n, 2/n) this set of particles moves following the sticky particle dynamics given also by a flow ϕ(t − (1/n), u0 , P (X1/n ∈ dx)), t ∈ [1/n, 2/n). By induction we repeat the same dynamics during the interval [k/n, (k + 1)/n) given by the flow ϕ(t − k/n, Xk/n , u0 , P (Xk/n ∈ dx)) with jumps at t = (k + 1)/n √ equal to ν/ nek+1 . The random variables (Xk/n : k ≥ 0) are given by induction as follows:
1 ν , Xk/n , u0 , P (Xk/n ∈ dx) + √ ek+1 , k ≥ 0. X(k+1)/n = ϕ n n By reiterating this dynamics, we construct a forward flow from R+ × Ω to Rd by
[nt] , X[nt]/n , u0 , P (X[nt]/n ∈ dx) . ψ(t, X0 , e) := ϕ t − n (n)
It follows from (6) that the process t → Xt := ψ(t, X0 , e), defined on (Ω, F, P ), satisfies t (n) (n) Xt = X0 + E[u0 (X0 )|Xs(n) ] ds + νBt ∀t ≥ 0, 0
where (n)
Bt (e) = n−1/2
[nt]
i=1
ei .
(7)
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4. Proof of the main results The proof of Theorem 2.2 is based on Itô’s formula. Before proving Theorem 2.3, we recall some results concerning the weak convergence on D([0, T ], Rd ). There is a metric dS , which we use here (see [1, pp. 112–116]) on D := D([0, T ], Rd ), with respect to which D is a complete separable metric space, and convergence of yn → y in the metric dS implies convergence at all points of continuity of y. Let D denote the Borel algebra induced on D by the metric dS . Let (Pn ) denote a tight sequence of probability measures on (D, D), corresponding to a sequence of processes Xn with path in D. Then (Pn ) has a weakly convergence subsequence to a probability measure P on (D, D), and there is a process X with paths in D corresponding to the probability P . If (Pn ) is tight we may write (X n ) is tight. The following lemma of Skorokhod [1] will be helpful later. Lemma 4.1. Let (vn ) and v denote random variables with values in a complete separable metric space (E, d). Then there exist random variables (v˜n ), v˜ with values in E such that for any Borel set A in E P (vn ∈ A) = P (v˜n ∈ A),
P (v ∈ A) = P (v˜ ∈ A).
The random variables (v˜n ), v˜ are defined on the same probability space, and d(v˜n , v) ˜ → 0 almost surely. (n)
Now, back to our proof of Theorem 2.3. Let X0 be any random variables with values in Rd , and X0 a sequence (n) of random variables with finite range such that X0 → X0 a.s. We have constructed (7), for each n, a process satisfying t (n) (n) (n) (n) Xt = X0 + E[u0 (X0 )|Xs(n) ] ds + νBt , t ≥ 0. 0
Now, the main idea of the proof is to show that this sequence of processes contains a subsequence that converges in (n) (n) D. Following a work of Kushner [6], there exists a subsequence of the processes (X· , B· ) that converges in D to the pair of continuous processes (X(ν), B), where Bt is a d-dimensional Brownian motion. Thanks to assumption (3), our limit process X(ν) satisfies t Xt (ν) = X0 + uν (Xs (ν), s) ds + νBt . 0
It only remains to prove that uν (Xt (ν), t) = E[u0 (X0 )|Xt (ν)]. Set, for t ≥ 0 (n)
Mt
(n)
(n)
:= E[u0 (X0 )|Xt ].
From Lemma 4.1, we can suppose that (n)
sup |Xt
t∈[0,T ]
− Xt (ν)| → 0
a.s.,
and then
t t (n) sup Ms ds − uν (Xs (ν), s) ds → 0
t∈[0,T ]
0
Let f, g be smooth functions. We have T (n) (n) E g(t)f (Xt )u0 (X0 ) dt = E 0
a.s.
0
0
T
(n)
g(t)f (Xt ) d
0
t
Ms(n) ds
.
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A. Dermoune, B. Djehiche / Physica D 163 (2002) 184–190
The fact that u0 (·) is continuous and bounded yields T T (n) (n) g(t)f (Xt )u0 (X0 ) dt → E g(t)f (Xt (ν))u0 (X0 ) dt , E and as
·
0
0
· converges uniformly in [0, T ] to 0 u(Xt (ν), t) ds, then
t
t T (n) (n) g(t)f (Xt ) d g(t)f (Xt (ν)) d Ms ds →E u(Xs (ν), s) ds .
(n) 0 Ms ds
T
E 0
0
0
Therefore, for all smooth functions f, g, T g(t)f (Xt (ν))u0 (X0 ) dt = E E 0
T
0
0
g(t)f (Xt (ν))u(Xt (ν), t) dt ,
and then E[u(Xt (ν), t)|Xt (ν)] = E[u0 (X0 )|Xt (ν)]P ⊗ dt
a.e.,
which yields that u(Xt (ν), t) = E[u0 (X0 )|Xt (ν)]. Proof of Theorem 2.4. It is easy to see that the sequence (X(ν), ν ∈ (0, 1]) is tight in C([0, T ]), and any limit as ν → 0 is solution of a differential equation dXt = ϕ(t) dt, where ϕ is some bounded process. The same proof as above shows that E[ϕ(t)|Xt ] = E[u0 (X0 )|Xt ] dt ⊗ P
a.e.
From the formula of change of variables we show that (P (Xt ∈ dx), u(x, t)) is a weak solution of the inviscid gas equations if and only if (4) is satisfied. If we suppose that this limit X is a Markov process σ (X0 )-measurable and the particles move on the real line, then [4] has shown that necessarily X represents the sticky particle dynamics studied in [2,7,8]. 䊐
Acknowledgements The authors acknowledge the suggestions of anonymous referees which significantly improved the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. Y. Brenier, E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal. 35 (6) (1998) 2317–2328. A. Dermoune, Probabilistic interpretation of sticky particle model, Ann. Probab. 27 (3) (1999) 1357–1367. A. Dermoune, d-Dimensional pressureless gas equations, Preprint, 2001. A. Dermoune, B. Djehiche, Pressureless gas equations with viscosity and nonlinear diffusions, CR Acad. Sci. Paris Sér. I 332 (2001) 745–750. E. Grenier, Existence globale pour le système des gaz sans pression, CR Acad. Sci. Paris Sér. I 321 (1995) 171–174. W. E, Y. Rykov, Y. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Commun. Math. Phys. 177 (1996) 349–380. H.J. Kushner, On the weak convergence of interpolated Markov chains to a diffusion, Ann. Probab. 2 (1) (1974) 40–50. M. Sever, An existence theorem in the large for zero-pressure gas dynamics, Diff. Integral Eqs., submitted for publication. S. Shandarin, Y. Zeldovich, The large scale structures of the universe: turbulence, intermittency, structures in a self-gravitating medium, Rev. Mod. Phys. 61 (1989) 185–220.
Global solution of the pressureless gas equation with viscosity Azzouz Dermoune a,∗ , Boualem Djehiche b a
Laboratoire de Probabilités et Statistique, UFR de Mathématiques, USTL, Bât. M2, 59655 Villeneuve d’Ascq Cédex, France b Department of Mathematics, KTH, 100 44 Stockholm, Sweden Received 9 July 2001; received in revised form 2 January 2002; accepted 7 January 2002 Communicated by U. Frisch
Abstract We construct a global weak solution to a d-dimensional system of zero-pressure gas dynamics modified by introducing a finite artificial viscosity. We use discrete approximations to the continuous gas and make particles move along trajectories of the normalized simple symmetric random walk with deterministic drift. The interaction of these particles is given by a sticky particle dynamics. We show that a subsequence of these approximations converges to a weak solution of the system of zero-pressure gas dynamics in the sense of distributions. This weak solution is interpreted in terms of a random process solution of a nonlinear stochastic differential equation. We get a weak solution of the inviscid system by tending the viscosity to zero. © 2002 Elsevier Science B.V. All rights reserved. PACS: 02.30.J; 02.30; 05.45; 05.40 Keywords: Pressureless gas equations with viscosity; Nonlinear diffusion process; Weak convergence
1. Introduction To explain the formation of large scale structures in the universe, Shandarin and Zeldovich [10] suggested the following toy model of sticky particles: consider a system of particles {xi0 } ⊂ Rd with initial velocities {vi0 } and masses {m0i }. The particles move with constant velocities unless they collide. At collision, the colliding particles stick and form a new massive particle. The laws of conservation of mass and momentum tell us that the probability distribution ρ(dx, t) and the velocity u(x, t) of this new particle are solutions of the following system of equations: ∂ρ + div(uρ) = 0, ∂t (S(0)) = ∂(uρ) + div(u ⊗ uρ) = 0, (∗) ∂t ρ(dx, t) → ρ0 (dx), u(x, t)ρ(dx, t) → u0 (x)ρ0 (dx), weakly as t → 0+ , ∗
Corresponding author. Tel.: +33-3-2043-6515; fax: +33-3-2043-6774. E-mail address: [email protected] (A. Dermoune). 0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 2 ) 0 0 3 5 4 - 8
A. Dermoune, B. Djehiche / Physica D 163 (2002) 184–190
185
called pressureless gas without viscosity. To solve this system of equations when d = 1, E et al. [7] devise a generalized variational principle while Dermoune [3] uses a probabilistic approach by constructing a stochastic process that satisfies the following nonlinear stochastic differential equation (SDE): dXt = E[u0 (X0 )|Xt ] dt,
L(X0 ) = ρ0 (dx).
In d-dimensional space, for any d > 1, the major difficulty of sticky particles is that point particles will generically move without collisions, thus generating a complicated multi-stream flow. In multidimensional case (see, e.g., [9]), the existence problem for zero-pressure gases is believed to be ill-posed. Nevertheless, in this paper, we will modify the Zeldovich model of sticky particles by assuming that the mass and momentum of the new particle have equal diffusion rates. Thus, the probability distribution ρ(dx, t) and the velocity u(x, t) of the new particle are weak solutions of the following system of gas equations with viscosity: ∂ρ ν2 + div(uρ) = ρ, ∂t 2 ν2 (S(ν)) = ∂(uρ) (∗∗ ) + div(u ⊗ uρ) = (uρ), ∂t 2 ρ(dx, t) → ρ0 (dx), u(x, t)ρ(dx, t) → u0 (x)ρ0 (dx), weakly as t → 0+ . In a previous paper (cf. [5]), we constructed a local strong solution of Eq. (∗∗) by solving the following nonlinear SDE: dXt = E[u0 (X0 )|Xt ] dt + ν dBt ,
L(X0 ) = ρ0 (dx),
(1)
where Bt is a standard Brownian motion independent of X0 , up to some time horizon T0 . This was done using a time reversal device of a forward diffusion process, which brings back the problem to the study of a system of quasi-parabolic equations whose solution exists only locally. The objective of the present work is in fact to construct a global weak solution of Eq. (∗∗) given any time horizon T , again by solving Eq. (1). The solution (ρ(dx, t), u(x, t)) of Eq. (∗∗) is then explicitly given as follows: ρ(dx, t) is the probability distribution of the diffusion process Xt at time t and the velocity vector is u(x, t) = E[u0 (X0 )|Xt = x]. In the simple case when X0 takes two values x1 , x2 , our nonlinear diffusion process given by Eq. (1) evolves as follows. Initially, we have two particles located at x1 and x2 , with masses m1 = P (X0 = x1 ), m2 = P (X0 = x2 ), and velocities v1 = u0 (x1 ), v2 = u0 (x2 ). Immediately each particle at xi diffuses with drift u(x, t) = α(x, t)v1 + (1 − α(x, t))v2 , where α(x, t) = P (X0 = x1 |Xt = x) is the proportion of diffusion trajectories starting from x1 and arriving to x at time t. (n) We approach the problem by constructing a system of particles whose positions (Xt , t ∈ [0, T ]) satisfy the following equation: t (n) (n) (n) (n) Xt = X0 + E[u0 (X0 )|Xs(n) ] ds + νBt , t ∈ [0, T ], (2) 0
186
A. Dermoune, B. Djehiche / Physica D 163 (2002) 184–190 (n)
(n)
where X0 is a random variable which has a finite range, and (Bt , t ∈ [0, T ]) is the properly normalized linear interpolation of the simple symmetric random walk on Zd . For each n, ν > 0 fixed we set (n)
(n)
un,ν (x, t) = E[u0 (X0 )|Xt
= x],
and suppose that for all ν > 0, there exists a function uν (x, t) continuous with respect to the variable x, dt a.e. and such that, for each M > 0, lim
sup |un,ν (x, t) − uν (x, t)| → 0,
n→+∞ |x|≤M
t > 0.
(3) (n)
By a standard limit theorem, we will show that the sequence (Xt , t ∈ [0, T ]) converges weakly in D([0, T ], Rd ), the space of functions on [0, T ] with values in Rd which are continuous from the right on [0, T ) and have left hand limits on (0, T ] to a process X(ν) satisfying Eq. (1). We will also show that any limit X of X(ν) as ν → 0 is solution of a differential equation dXt = ϕ(t) dt. If ρ(dx, t) = P (Xt ∈ dx) denotes the probability distribution of Xt and u(x, t) = E[ϕ(t)|Xt = x], then (ρ(dx, t), u(x, t)) is solution of the inviscid gas equations if and only if for every 1 ≤ j ≤ d, E[ϕj (t)u0 (X0 )|Xt ] = E[ϕj (t)|Xt ]E[u0 (X0 )|Xt ] dt ⊗ P
a.e.
(4)
Here ϕj (t) are the components of the vector ϕ(t). In Section 2, we state the main results of the paper. Our system of sticky particles is discussed in Section 3. Finally, in Section 4, we prove the main result, by characterizing the limit process of the system of sticky particles, as solution to Eq. (1).
2. Main results We will denote by ·, · the scalar product in the Euclidean space Rd . By a weak solution of Eq. (∗∗), we mean the following definition. Definition 2.1. A family (ρ(dx, t), u(x, t)ρ(dx, t), t ∈ [0, T ]) of Borel measures is a weak solution of Eq. (∗∗), if for any f ∈ C02 , the space of C 2 -functions with compact support, and any 0 < t1 < t2 : t t (D1) f (x)ρ(dx, t2 ) − f (x)ρ(dx, t1 ) = t12 ∇f (x), u(x, t)ρ(dx, t) dt + (ν 2 /2) t12 f (x)ρ(dx, t) dt; t t (D2) f (x)u(x, t2 )ρ(dx, t2 ) − f (x)u(x, t1 )ρ(dx, t1 ) = (ν 2 /2) t12 f (x)u(x, t)ρ(dx, t) dt + t12 ∇f (x), u(x, t)u(x, t)ρ(dx, t) dt; (D3) ρ(dx, t) → ρ0 (dx), u(x, t)ρ(dx, t) → u0 (x)ρ0 (dx) weakly as t → 0+ . Now we state the main result of the paper. Theorem 2.2. Let X be any solution of dXt = E[u0 (X0 )|Xt ] + ν dBt . We denote by ρ(dx, t, ν) the probability distribution of Xt , and u(x, t, ν) = E[u0 (X0 )|Xt = x]. Then (ρ(dx, t, ν), u(x, t, ν)ρ(dx, t, ν), t ∈ [0, T ]) is a weak solution of Eq. (∗∗).
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Theorem 2.3. Assume that x → u0 (x) is bounded, continuous and ρ0 (dx) a probability distribution on Rd . Under the assumption (3), for all T > 0, there exists a probability measure on C([0, T ], Rd ) under which the coordinate process Xt satisfies dXt = E[u0 (X0 )|Xt ] dt + ν dBt ,
t ∈ [0, T ],
L(X0 ) = ρ0 (dx),
where B is a d-dimensional Brownian motion. Theorem 2.4. Under the assumption (4), there exists a sequence νn → 0 such that the sequence (ρ(dx, t, νn ), u(x, t, νn )ρ(dx, t, νn )) converges weakly to (ρ(dx, t, 0), u(x, t, 0)ρ(dx, t, 0)) solution of zero-pressure gas equations, Eq. (∗), without viscosity.
3. A system of sticky particles 0 } ⊂ R d , with initial velocities Let us first study a system Σ of N particles initially located at {x10 , . . . , xN 0 0 0 u0 (x1 ), . . . , u0 (xN ). Each particle xi has the mass mi . Sticky particles dynamics for the system Σ can be described by the following system of ordinary differential equations:
lim
ε→0+
Xt+ε − Xt = f (Xt ), ε
t > 0,
(5)
where f is the map from RNd to RNd defined by N j =1 mj u0 (xj )1[xj =xi ] fi (X) = , 1 ≤ i ≤ N, N j =1 mj 1[xj =xi ] where f = (f1 , . . . , fN ), X = (x1 , . . . , xN ) are the coordinates, respectively, of f and X. We show easily that for each X0 ∈ RNd the system (5) has a unique solution which starts from X0 . If we denote by t → ϕ(t, X0 ) the latter solution, then the map ϕ : R+ × RNd → RNd satisfies the flow property: (1) ϕ(0, X0 ) = X0 , (2) ϕ(t + s, X0 ) = ϕ(t, ϕ(s, X0 )), s, t ≥ 0. 0 ), then ϕ(t, X ) is the location of the system Σ at time t. The particle, initially located If we take X0 = (x10 , . . . , xN 0 0 at xi , is located at ϕi (t, X0 ) at time t. Now we give a probabilistic interpretation of the flow ϕ. Suppose that N i=1 mi = 1, and consider the proba 0 ). Each random variable X defined from the bility space (Rd , B(Rd ), ρ0 (dx)), where ρ0 (dx) = N m δ(x − x i i=1 i 0 )) ∈ R Nd . Using probability space (Rd , B(Rd ), ρ0 (dx)) to Rd can be seen as a vector X := (X(x10 ), . . . , X(xN 0 0 this identification the vectors X0 = (x1 , . . . , xN ), ϕ(t, X0 ) = (ϕ1 (t, X0 ), . . . , ϕN (t, X0 )) represent, respectively, the random variables X0 : xi0 ∈ Rd → xi0 , and xi0 ∈ Rd → ϕi (t, X0 ). It follows from (5) that the process t → ϕ(t, X0 ) := Xt satisfies the following SDE:
dXt = E[u0 (X0 )|Xt ], dt
(6)
where E[·|Xt ] is the conditional expectation with respect to the random variable Xt . Remark that the flow ϕ(t, X0 ) := ϕ(t, X0 , u0 , ρ0 ) depends only on the velocity u0 and the probability distribution ρ0 (dx) := P (X0 ∈ dx) of the random variable X0 . We will adopt this notation in the sequel.
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Now we come back to the construction of (2). Before doing that we need some notations. Let Ω0 := ({−1, 1}d )N be the set of sequences (en : n ≥ 1) with values in {−1, +1}d . The σ -algebra F0 is spanned by the coordinate (e ∈ Ω0 → πn (e) = en : n ≥ 1). The space (Ω0 , F0 ) is endowed with the probability ⊗N
1 . µ= d 2 d e∈{−1,1}
We consider the product space Ω := Rd × Ω0 , endowed with the σ -algebra F generated by the set B × A for B ∈ B(Rd ) and A ∈ F0 , and the probability measure P := ρ0 (dx) ⊗ µ. Let us construct for each integer n a process satisfying (2) in the probability space (Ω, F, P ). We consider a system of particles with initial locations {xi0 : 1 ≤ i ≤ N } which evolves according to the following dynamics. At time zero the particles are located at (xi0 ), with assigned velocities or drifts (u0 (xi0 )), and masses (mi ). During the interval of time [0, 1/n) this set of particles moves following the sticky particle dynamics given above by the flow ϕ(t, X0 , u0 , P (X0 ∈ dx)). At time t = 1/n each particle ϕi (1/n, X0 , u0 , P (X0 ∈ dx)) jumps to √ ϕi (1/n, X0 , u0 , P (X0 ∈ dx)) + ν/ ne1 , and we get a new system of particles X1/n := ϕ(1/n, X0 , u0 , P (X0 ∈ √ dx)) + ν/ ne1 . At each location x = X1/n there is a massive particle with mass m(x) = P (X1/n = x), and velocity
1 = E[u0 (X0 )|X1/n = x]. u x, n Now again during the interval [1/n, 2/n) this set of particles moves following the sticky particle dynamics given also by a flow ϕ(t − (1/n), u0 , P (X1/n ∈ dx)), t ∈ [1/n, 2/n). By induction we repeat the same dynamics during the interval [k/n, (k + 1)/n) given by the flow ϕ(t − k/n, Xk/n , u0 , P (Xk/n ∈ dx)) with jumps at t = (k + 1)/n √ equal to ν/ nek+1 . The random variables (Xk/n : k ≥ 0) are given by induction as follows:
1 ν , Xk/n , u0 , P (Xk/n ∈ dx) + √ ek+1 , k ≥ 0. X(k+1)/n = ϕ n n By reiterating this dynamics, we construct a forward flow from R+ × Ω to Rd by
[nt] , X[nt]/n , u0 , P (X[nt]/n ∈ dx) . ψ(t, X0 , e) := ϕ t − n (n)
It follows from (6) that the process t → Xt := ψ(t, X0 , e), defined on (Ω, F, P ), satisfies t (n) (n) Xt = X0 + E[u0 (X0 )|Xs(n) ] ds + νBt ∀t ≥ 0, 0
where (n)
Bt (e) = n−1/2
[nt]
i=1
ei .
(7)
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4. Proof of the main results The proof of Theorem 2.2 is based on Itô’s formula. Before proving Theorem 2.3, we recall some results concerning the weak convergence on D([0, T ], Rd ). There is a metric dS , which we use here (see [1, pp. 112–116]) on D := D([0, T ], Rd ), with respect to which D is a complete separable metric space, and convergence of yn → y in the metric dS implies convergence at all points of continuity of y. Let D denote the Borel algebra induced on D by the metric dS . Let (Pn ) denote a tight sequence of probability measures on (D, D), corresponding to a sequence of processes Xn with path in D. Then (Pn ) has a weakly convergence subsequence to a probability measure P on (D, D), and there is a process X with paths in D corresponding to the probability P . If (Pn ) is tight we may write (X n ) is tight. The following lemma of Skorokhod [1] will be helpful later. Lemma 4.1. Let (vn ) and v denote random variables with values in a complete separable metric space (E, d). Then there exist random variables (v˜n ), v˜ with values in E such that for any Borel set A in E P (vn ∈ A) = P (v˜n ∈ A),
P (v ∈ A) = P (v˜ ∈ A).
The random variables (v˜n ), v˜ are defined on the same probability space, and d(v˜n , v) ˜ → 0 almost surely. (n)
Now, back to our proof of Theorem 2.3. Let X0 be any random variables with values in Rd , and X0 a sequence (n) of random variables with finite range such that X0 → X0 a.s. We have constructed (7), for each n, a process satisfying t (n) (n) (n) (n) Xt = X0 + E[u0 (X0 )|Xs(n) ] ds + νBt , t ≥ 0. 0
Now, the main idea of the proof is to show that this sequence of processes contains a subsequence that converges in (n) (n) D. Following a work of Kushner [6], there exists a subsequence of the processes (X· , B· ) that converges in D to the pair of continuous processes (X(ν), B), where Bt is a d-dimensional Brownian motion. Thanks to assumption (3), our limit process X(ν) satisfies t Xt (ν) = X0 + uν (Xs (ν), s) ds + νBt . 0
It only remains to prove that uν (Xt (ν), t) = E[u0 (X0 )|Xt (ν)]. Set, for t ≥ 0 (n)
Mt
(n)
(n)
:= E[u0 (X0 )|Xt ].
From Lemma 4.1, we can suppose that (n)
sup |Xt
t∈[0,T ]
− Xt (ν)| → 0
a.s.,
and then
t t (n) sup Ms ds − uν (Xs (ν), s) ds → 0
t∈[0,T ]
0
Let f, g be smooth functions. We have T (n) (n) E g(t)f (Xt )u0 (X0 ) dt = E 0
a.s.
0
0
T
(n)
g(t)f (Xt ) d
0
t
Ms(n) ds
.
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The fact that u0 (·) is continuous and bounded yields T T (n) (n) g(t)f (Xt )u0 (X0 ) dt → E g(t)f (Xt (ν))u0 (X0 ) dt , E and as
·
0
0
· converges uniformly in [0, T ] to 0 u(Xt (ν), t) ds, then
t
t T (n) (n) g(t)f (Xt ) d g(t)f (Xt (ν)) d Ms ds →E u(Xs (ν), s) ds .
(n) 0 Ms ds
T
E 0
0
0
Therefore, for all smooth functions f, g, T g(t)f (Xt (ν))u0 (X0 ) dt = E E 0
T
0
0
g(t)f (Xt (ν))u(Xt (ν), t) dt ,
and then E[u(Xt (ν), t)|Xt (ν)] = E[u0 (X0 )|Xt (ν)]P ⊗ dt
a.e.,
which yields that u(Xt (ν), t) = E[u0 (X0 )|Xt (ν)]. Proof of Theorem 2.4. It is easy to see that the sequence (X(ν), ν ∈ (0, 1]) is tight in C([0, T ]), and any limit as ν → 0 is solution of a differential equation dXt = ϕ(t) dt, where ϕ is some bounded process. The same proof as above shows that E[ϕ(t)|Xt ] = E[u0 (X0 )|Xt ] dt ⊗ P
a.e.
From the formula of change of variables we show that (P (Xt ∈ dx), u(x, t)) is a weak solution of the inviscid gas equations if and only if (4) is satisfied. If we suppose that this limit X is a Markov process σ (X0 )-measurable and the particles move on the real line, then [4] has shown that necessarily X represents the sticky particle dynamics studied in [2,7,8]. 䊐
Acknowledgements The authors acknowledge the suggestions of anonymous referees which significantly improved the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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