Comptes Rendus
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+ 0 t u(X s a ,s)ds+νB t a ,X t b =b+ 0 t u(X s b ,s)ds+νB 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 a,b,v 1 ,v 2 R d and p∈(0,1) are given. We show that (ρ t (x)=pρ t a (x)+(1-p)ρ t b (x),u(x,t):t0,xR d ) is the unique weak solution of the following pressureless gas system

𝒮(d,ν){ t (ρ)+ j=1 d x j (u j ρ)=ν 2 2Δ(ρ), t (u i ρ)+ j=1 d x j (u i u j ρ)=ν 2 2Δ(u i ρ),1id,
such that ρ t (x)dxpδ a +(1-p)δ b ,u(x,t)ρ t (x)dx pv 1 δ a +(1-p)v 2 δ b as t→0+.

Nous construisons deux diffusions indépendantes X t a =a+ 0 t u(X s a ,s)ds+νB t a ,X t b =b+ 0 t u(X s b ,s)ds+νB 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 ρ t a ,ρ t b sont respectivement les densités de Xta et Xtb. Ici a,b,v 1 ,v 2 R d , et p∈(0,1) sont donnés. Nous montrons que la famille (ρ t (x)=pρ t a (x)+(1-p)ρ t b (x),u(x,t):t0,xR d ) est l'unique solution faible du système de gaz sans pression 𝒮(d,ν) cité dans l'Abstract.

Received:
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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     title = {Diffusion with interactions between two types of particles and {Pressureless} gas equations},
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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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