Diffusion with interactions between two types of particles and Pressureless gas equations

Dermoune, Azzouz; Filali, Siham

doi:10.1016/j.crma.2003.10.018

Probability Theory

Diffusion with interactions between two types of particles and Pressureless gas equations
[Diffusion avec interaction entre deux types de particules et système de gaz sans pression avec viscosité]

Présenté par : Marc Yor

Dermoune, Azzouz ¹ ; Filali, Siham ¹

¹ Laboratoire de mathématiques appliquées, équipe de probabilités et statistique, F.R.E. 2222, UFR de mathématiques, USTL, bât. M2, 59655 Villeneuve d'Ascq cédex, France

Comptes Rendus. Mathématique, Tome 337 (2003) no. 11, pp. 731-735.

Résumé
Abstract

Nous construisons deux diffusions indépendantes $X_{t}^{a} = a + \int_{0}^{t} u (X_{s}^{a}, s) d s + ν B_{t}^{a}, X_{t}^{b} = b + \int_{0}^{t} u (X_{s}^{b}, s) d s + ν B_{t}^{b}$ , ayant la même viscosité ν≠0 et la même dérive u(x,t)=(pρ_t^a(x)v₁+(1−p)ρ_t^b(x)v₂)/(pρ_t^a(x)+(1−p)ρ_t^b(x)), où $ρ_{t}^{a}, ρ_{t}^{b}$ sont respectivement les densités de X_t^a et X_t^b. Ici $a, b, v_{1}, v_{2} \in R^{d}$ , et p∈(0,1) sont donnés. Nous montrons que la famille $(ρ_{t} (x) = p ρ_{t}^{a} (x) + (1 - p) ρ_{t}^{b} (x), u (x, t) : t ⩾ 0, x \in R^{d})$ est l'unique solution faible du système de gaz sans pression $𝒮 (d, ν)$ cité dans l'Abstract.

We construct two d-dimensional independent diffusions $X_{t}^{a} = a + \int_{0}^{t} u (X_{s}^{a}, s) d s + ν B_{t}^{a}, X_{t}^{b} = b + \int_{0}^{t} u (X_{s}^{b}, s) d s + ν B_{t}^{b}$ , with the same viscosity ν≠0 and the same drift u(x,t)=(pρ_t^a(x)v₁+(1−p)ρ_t^b(x)v₂)/(pρ_t^a(x)+(1−p)ρ_t^b(x)), where ρ_t^a,ρ_t^b are respectively the density of X_t^a and X_t^b. Here $a, b, v_{1}, v_{2} \in R^{d}$ and p∈(0,1) are given. We show that $(ρ_{t} (x) = p ρ_{t}^{a} (x) + (1 - p) ρ_{t}^{b} (x), u (x, t) : t ⩾ 0, x \in R^{d})$ is the unique weak solution of the following pressureless gas system

𝒮 (d, ν) {\begin{matrix} \partial_{t} (ρ) + \sum_{j = 1}^{d} \partial_{x_{j}} (u_{j} ρ) = \frac{ν^{2}}{2} Δ (ρ), \\ \partial_{t} (u_{i} ρ) + \sum_{j = 1}^{d} \partial_{x_{j}} (u_{i} u_{j} ρ) = \frac{ν^{2}}{2} Δ (u_{i} ρ), \forall 1 ⩽ i ⩽ d, \end{matrix}

such that

ρ_{t} (x) d x \to p δ_{a} + (1 - p) δ_{b}, u (x, t) ρ_{t} (x) d x \to {pv}_{1} δ_{a} + (1 - p) v_{2} δ_{b}

as t→0+.

Reçu le : 2003-05-27
Accepté le : 2003-10-14
Publié le : 2003-11-14

DOI : 10.1016/j.crma.2003.10.018

Affiliations des auteurs :

Dermoune, Azzouz ¹ ; Filali, Siham ¹

¹ Laboratoire de mathématiques appliquées, équipe de probabilités et statistique, F.R.E. 2222, UFR de mathématiques, USTL, bât. M2, 59655 Villeneuve d'Ascq cédex, France

@article{CRMATH_2003__337_11_731_0,
     author = {Dermoune, Azzouz and Filali, Siham},
     title = {Diffusion with interactions between two types of particles and {Pressureless} gas equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {731--735},
     publisher = {Elsevier},
     volume = {337},
     number = {11},
     year = {2003},
     doi = {10.1016/j.crma.2003.10.018},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2003.10.018/}
}

TY  - JOUR
AU  - Dermoune, Azzouz
AU  - Filali, Siham
TI  - Diffusion with interactions between two types of particles and Pressureless gas equations
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 731
EP  - 735
VL  - 337
IS  - 11
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2003.10.018/
DO  - 10.1016/j.crma.2003.10.018
LA  - en
ID  - CRMATH_2003__337_11_731_0
ER  -

%0 Journal Article
%A Dermoune, Azzouz
%A Filali, Siham
%T Diffusion with interactions between two types of particles and Pressureless gas equations
%J Comptes Rendus. Mathématique
%D 2003
%P 731-735
%V 337
%N 11
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2003.10.018/
%R 10.1016/j.crma.2003.10.018
%G en
%F CRMATH_2003__337_11_731_0

Dermoune, Azzouz; Filali, Siham. Diffusion with interactions between two types of particles and Pressureless gas equations. Comptes Rendus. Mathématique, Tome 337 (2003) no. 11, pp. 731-735. doi : 10.1016/j.crma.2003.10.018. http://www.numdam.org/articles/10.1016/j.crma.2003.10.018/

Bibliographie
Cité par

[1] Brenier, Y.; Grenier, E. Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., Volume 35 (1998), pp. 2317-2328

[2] Chow, Y.S.; Teicher, H. Probability Theory, Springer, New York, 1978

[3] Dermoune, A. Propagation of chaos for pressureless gas equations, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 935-940

[4] Dermoune, A. Propagation and conditional propagation of chaos for pressureless gas equations, Probab. Theory Related Fields, Volume 126 (2003), pp. 459-476

[5] Dermoune, A.; Djehiche, B. Pressureless gas equations with viscosity and nonlinear diffusion, C. R. Acad. Sci. Paris, Ser. I, Volume 332 (2001), pp. 741-750

[6] Dermoune, A.; Djehiche, B. Global solution of pressureless gas equation with viscosity, Physica D, Volume 163 (2002), pp. 184-190

[7] Weinan, E.; Rykov, Y.; Sinai, Y. Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion dynamics, Comm. Math. Phys., Volume 177 (1996), pp. 349-380

[8] Sever, M. An existence theorem in the large for zero-pressure gas dynamics, Differential Integral Equations, Volume 14 (2001) no. 9, pp. 1077-1092

[9] Zheng, W. Conditional propagation of chaos and a class of quasilinear PDE's, Ann. Probab., Volume 23 (1995) no. 3, pp. 1389-1413

Cité par Sources :