Probability Theory
Diffusion with interactions between two types of particles and Pressureless gas equations
Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 731-735.

We construct two d-dimensional independent diffusions ${X}_{t}^{a}=a+{\int }_{0}^{t}u\left({X}_{s}^{a},s\right)\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}s+\nu {B}_{t}^{a},\phantom{\rule{3.30002pt}{0ex}}{X}_{t}^{b}=b+{\int }_{0}^{t}u\left({X}_{s}^{b},s\right)\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}s+\nu {B}_{t}^{b}$, with the same viscosity ν≠0 and the same drift u(x,t)=(ta(x)v1+(1−p)ρtb(x)v2)/(ta(x)+(1−p)ρtb(x)), where ρta,ρtb are respectively the density of Xta and Xtb. Here $\mathrm{a},\mathrm{b},{\mathrm{v}}_{1},{\mathrm{v}}_{2}\in {\mathrm{R}}^{d}$ and p∈(0,1) are given. We show that $\left({\rho }_{t}\left(\mathrm{x}\right)=\mathrm{p}{\rho }_{t}^{a}\left(\mathrm{x}\right)+\left(1-\mathrm{p}\right){\rho }_{t}^{b}\left(\mathrm{x}\right),\mathrm{u}\left(\mathrm{x},\mathrm{t}\right):\phantom{\rule{3.30002pt}{0ex}}\mathrm{t}⩾0,\phantom{\rule{3.30002pt}{0ex}}\mathrm{x}\in {\mathrm{R}}^{d}\right)$ is the unique weak solution of the following pressureless gas system

 $𝒮\left(d,\nu \right)\phantom{\rule{10.0pt}{0ex}}\left\{\begin{array}{c}{\partial }_{t}\left(\rho \right)+\sum _{j=1}^{d}{\partial }_{{x}_{j}}\left({u}_{j}\rho \right)=\frac{{\nu }^{2}}{2}\Delta \left(\rho \right),\hfill \\ {\partial }_{t}\left({u}_{i}\rho \right)+\sum _{j=1}^{d}{\partial }_{{x}_{j}}\left({u}_{i}{u}_{j}\rho \right)=\frac{{\nu }^{2}}{2}\Delta \left({u}_{i}\rho \right),\phantom{\rule{10.0pt}{0ex}}\forall 1⩽i⩽d,\hfill \end{array}$
such that ${\rho }_{t}\left(\mathrm{x}\right)\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}\mathrm{x}\to \mathrm{p}{\delta }_{a}+\left(1-\mathrm{p}\right){\delta }_{b},\phantom{\rule{3.30002pt}{0ex}}\mathrm{u}\left(\mathrm{x},\mathrm{t}\right){\rho }_{t}\left(\mathrm{x}\right)\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}\mathrm{x}\to {\mathrm{pv}}_{1}{\delta }_{a}+\left(1-\mathrm{p}\right){\mathrm{v}}_{2}{\delta }_{b}$ as t→0+.

Nous construisons deux diffusions indépendantes ${X}_{t}^{a}=a+{\int }_{0}^{t}u\left({X}_{s}^{a},s\right)\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}s+\nu {B}_{t}^{a},\phantom{\rule{3.30002pt}{0ex}}{X}_{t}^{b}=b+{\int }_{0}^{t}u\left({X}_{s}^{b},s\right)\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}s+\nu {B}_{t}^{b}$, ayant la même viscosité ν≠0 et la même dérive u(x,t)=(ta(x)v1+(1−p)ρtb(x)v2)/(ta(x)+(1−p)ρtb(x)), où ${\rho }_{t}^{a},\phantom{\rule{3.30002pt}{0ex}}{\rho }_{t}^{b}$ sont respectivement les densités de Xta et Xtb. Ici $\mathrm{a},\mathrm{b},{\mathrm{v}}_{1},{\mathrm{v}}_{2}\in {\mathrm{R}}^{d}$, et p∈(0,1) sont donnés. Nous montrons que la famille $\left({\rho }_{t}\left(\mathrm{x}\right)=\mathrm{p}{\rho }_{t}^{a}\left(\mathrm{x}\right)+\left(1-\mathrm{p}\right){\rho }_{t}^{b}\left(\mathrm{x}\right),\mathrm{u}\left(\mathrm{x},\mathrm{t}\right):\phantom{\rule{3.30002pt}{0ex}}\mathrm{t}⩾0,\phantom{\rule{3.30002pt}{0ex}}\mathrm{x}\in {\mathrm{R}}^{d}\right)$ est l'unique solution faible du système de gaz sans pression $𝒮\left(\mathrm{d},\nu \right)$ cité dans l'Abstract.

Accepted:
Published online:
DOI: 10.1016/j.crma.2003.10.018
Azzouz Dermoune 1; Siham Filali 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
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Azzouz Dermoune; Siham Filali. Diffusion with interactions between two types of particles and Pressureless gas equations. Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 731-735. doi : 10.1016/j.crma.2003.10.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.018/

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