# Preprint serie A 153/2002

Asymptotic stability of Riemann solutions for a class of multi-D viscous systems of conservation laws.

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Keywords:
asymptotic stability, Riemann solutions, conservation laws, multi-D problems

Abstract:
We prove the asymptotic stability of two-states nonplanar Riemann solutions under initial and viscous perturbations for a class of multidimensional systems of conservation laws. The class of systems to which our result applies is constituted by those systems whose flux-functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. In particular, we obtain the uniqueness of the self-similar $L^\infty$ entropy solution of the two-states nonplanar Riemann problem. 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 $\xix=\xx/t$. That is, the solution $u(t,\xx)$ of the perturbed problem satisfies $u(t,t\xix)\to R(\xix)$ as $t\to\infty$, in $L_{\loc}^1(\R^n)$, where $R(\xix)$ is the self-similar entropy solution of the corresponding two-states nonplanar Riemann problem.

MSC 2000:
35B35    Stability, boundedness
35B40    Asymptotic behavior of solutions