Preprint A218/2003
Asymptotic Stability of Non-Planar Riemann Solutions for a Special Class of Multi-D Systems of Conservation Laws

Hermano Frid

**Keywords: **
Conservation laws | entropy solutions | Riemann problems | asymptotic stability | multidimensional systems

We study the asymptotic behavior of entropy solutions of the Cauchy problem for multidimensional systems of conservation laws of the form $\partial_tu+\sum_{\alpha=1}^d\partial_\alpha(g^\alpha(|u|)\,u)=0$, where the $g^\alpha$ are real smooth functions defined in $[0,+\infty)$, when the initial data are perturbations of two states nonplanar Riemann data. Specifically, if $R_0(x)$ is such Riemann data and $\psi\in L^\infty(\R^d)^n$ satisfies $\psi(Tx)\to0$ in $L_{loc}^1(\R^d)^n$, as $T\to\infty$, then an entropy solution, $u(x,t)$, of the Cauchy problem with $u(x,0)=R_0(x)+\psi(x)$ satisfies
$u(\xi t,t)\to R(\xi)$ in $L_{loc}^1(\R^d)^n$, as $t\to\infty$, where $R(x/t)$ turns out to be the unique self-similar entropy solution of the corresponding Riemann problem.