### Asymptotic Behavior of Smooth Solutions for Partially Dissipative Hyberbolic Systems with a Convex Entropy

S. Bianchini, B. Hanouzet, and R. Natalini

Abstract: We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in $m$ space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach constant equilibrium state in the $L^p$-norm at a rate $O(t^{-\frac{m}{2}(1-\frac{1}{p})})$, as $t\to\infty$, for $p\in [\min{\{m,2\}},\infty]$. Moreover, we can show that we can approximate, with a faster order of convergence, the conservative part of the solution in terms of the linearized hyperbolic operator for $m\geq 2$, and by a parabolic equation, in the spirit of Chapman-Enskog expansion. The main tool is given by a detailed analysis of the Green function for the linearized problem.

Paper:
Available as PDF (392 Kbytes).
Author(s):
S. Bianchini, <bianchin@sissa.it>
B. Hanouzet, <Bernard.Hanouzet@math.u-bordeaux1.fr>
R. Natalini, <r.natalini@iac.cnr.it>
Publishing information:
IAC Report 79 (11/2005)