Propagation of chaos for pressureless gas equations with viscosity
Comptes Rendus. Mathématique, Volume 335 (2002) no. 11, pp. 935-940.

We use A.S. Sznitman ideas of probabilistic phenomenon of propagation of chaos for Burgers equation, and we derive the existence and uniqueness of a weak solution of the following system of pressureless gas equations with viscosity:

 $\left(𝒮\right)\phantom{\rule{10.0pt}{0ex}}\left\{\begin{array}{c}\frac{\partial }{\partial t}\rho +\frac{\partial }{\partial x}\left(u\rho \right)=\frac{1}{2}\frac{{\partial }^{2}}{{\partial }^{2}x}\rho ,\hfill \\ \frac{\partial }{\partial t}\left(u\rho \right)+\frac{\partial }{\partial x}\left({u}^{2}\rho \right)=\frac{1}{2}\frac{{\partial }^{2}}{{\partial }^{2}x}\left(u\rho \right),\hfill \\ \rho \left(\mathrm{d}x,t\right)\to \rho \left(\mathrm{d}x,0\right),u\left(x,t\right)\rho \left(\mathrm{d}x,t\right)\to {u}_{0}\left(x\right)\rho \left(\mathrm{d}x,0\right)\phantom{\rule{10.0pt}{0ex}}\mathrm{weakly}\phantom{\rule{0.277778em}{0ex}}\mathrm{as}\phantom{\rule{3.30002pt}{0ex}}t\to {0}^{+}.\hfill \end{array}$

Dans cette Note on utilise les idées de A.S. Sznitman dans son étude de la propagation du chaos probabiliste pour l'équation de Burgers, et on obtient l'existence et l'unicité d'une solution faible au système $\left(𝒮\right)$ de gaz sans pression avec viscosité cité dans l'abstract.

Accepted:
Published online:
DOI: 10.1016/S1631-073X(02)02602-X
Azzouz Dermoune 1

1 Laboratoire de probabilités et statistique, UFR de mathématiques, USTL, bât. M2, 59655 Villeneuve d'Ascq cedex, France
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Azzouz Dermoune. Propagation of chaos for pressureless gas equations with viscosity. Comptes Rendus. Mathématique, Volume 335 (2002) no. 11, pp. 935-940. doi : 10.1016/S1631-073X(02)02602-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02602-X/

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