Système de gaz sans pression avec viscosité et diffusion non linéaire

Azzouz Dermoune; Boualem Djehiche

doi:10.1016/S0764-4442(01)01914-0

Système de gaz sans pression avec viscosité et diffusion non linéaire

Translated title of the contribution: Pressureless gas equations with viscosity and nonlinear diffusion

Azzouz Dermoune, Boualem Djehiche

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Under some regularity conditions on P₀ and u₀, we derive a unique local strong solution of the following system of pressureless gas equations with viscosity: ∂P_t∂t+div(uP_t)=σ²2ΔP _t,∂(uP_t)∂t+div(u²P _t)=σ²2Δ(uP_t),P _t→P₀,uP_t→u₀P ₀,weakly, ast→0⁺, by constructing a nonlinear diffusion process as solution to the following SDE: X_t=X₀+∫0tE[u₀(X₀)X _s]ds+σB_t,L(X₀)=P₀. We show then that P_t is the probability density of X_t while the velocity field admits the following stochastic representation: u(t,x)=E[u₀(X₀)X_t=x].

Translated title of the contribution	Pressureless gas equations with viscosity and nonlinear diffusion
Original language	French
Pages (from-to)	745-750
Number of pages	6
Journal	Comptes Rendus de l'Academie des Sciences - Series I: Mathematics
Volume	332
Issue number	8
DOIs	https://doi.org/10.1016/S0764-4442(01)01914-0
State	Published - Apr 15 2001

ASJC Scopus subject areas

General Mathematics

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10.1016/S0764-4442(01)01914-0

Cite this

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title = "Syst{\`e}me de gaz sans pression avec viscosit{\'e} et diffusion non lin{\'e}aire",

abstract = "Under some regularity conditions on P0 and u0, we derive a unique local strong solution of the following system of pressureless gas equations with viscosity: ∂Pt∂t+div(uPt)=σ22ΔP t,∂(uPt)∂t+div(u2P t)=σ22Δ(uPt),P t→P0,uPt→u0P 0,weakly, ast→0+, by constructing a nonlinear diffusion process as solution to the following SDE: Xt=X0+∫0tE[u0(X0)X s]ds+σBt,L(X0)=P0. We show then that Pt is the probability density of Xt while the velocity field admits the following stochastic representation: u(t,x)=E[u0(X0)Xt=x].",

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T1 - Système de gaz sans pression avec viscosité et diffusion non linéaire

AU - Dermoune, Azzouz

AU - Djehiche, Boualem

PY - 2001/4/15

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N2 - Under some regularity conditions on P0 and u0, we derive a unique local strong solution of the following system of pressureless gas equations with viscosity: ∂Pt∂t+div(uPt)=σ22ΔP t,∂(uPt)∂t+div(u2P t)=σ22Δ(uPt),P t→P0,uPt→u0P 0,weakly, ast→0+, by constructing a nonlinear diffusion process as solution to the following SDE: Xt=X0+∫0tE[u0(X0)X s]ds+σBt,L(X0)=P0. We show then that Pt is the probability density of Xt while the velocity field admits the following stochastic representation: u(t,x)=E[u0(X0)Xt=x].

AB - Under some regularity conditions on P0 and u0, we derive a unique local strong solution of the following system of pressureless gas equations with viscosity: ∂Pt∂t+div(uPt)=σ22ΔP t,∂(uPt)∂t+div(u2P t)=σ22Δ(uPt),P t→P0,uPt→u0P 0,weakly, ast→0+, by constructing a nonlinear diffusion process as solution to the following SDE: Xt=X0+∫0tE[u0(X0)X s]ds+σBt,L(X0)=P0. We show then that Pt is the probability density of Xt while the velocity field admits the following stochastic representation: u(t,x)=E[u0(X0)Xt=x].

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ER -

Système de gaz sans pression avec viscosité et diffusion non linéaire

Abstract

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