# Preprint serie A 381/2005

Asymptotic stability of Riemann solutions in BGK approximations to certain multidimensional systems of conservation laws

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Keywords:
conservation laws, BGK approximations,Riemann problems, asymptotic stability, multidimensional systems

Abstract:
We prove the asymptotic stability of nonplanar two-states Riemann solutions in BGK approximations of a class of multidimensional systems of conservation laws. The latter consists of systems whose flux-functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. The asymptotic stability to which the main result refers is in the sense of the convergence as $t\to\infty$ in $L_{\loc}^1$ of the space of directions $\bd=\xx/t$. That is, the solution $z(t,\xx,\xi)$ of the perturbed Cauchy problem for the corresponding BGK system satisfies $\int_X z(t,t\bd,\xi)d\mu(\xi)\to R(\bd)$ as $t\to\infty$, in $L_{\loc}^1(\R^n)$, where $R(\bd)$ is the self-similar entropy solution of the two-states nonplanar Riemann problem for the system of conservation laws.

MSC 2000:
35B40    Asymptotic behavior of solutions
35B35    Stability, boundedness
35L65    Conservation laws
35K55    Nonlinear PDE of parabolic type