Asymptotic Behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

摘要

We study the asymptotic time behavior of global smooth solutions to general entropy, dissipative, hyperbolic systems of balance laws in m space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach a constant equilibrium state in the L-P-norm at a rate O(t(-(m/2)(1-1/P))) as t --> infinity for p epsilon [min {m, 2}, infinity]. 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 >= 2, and by a parabolic equation, in the spirit of Chapman-Enskog expansion in every space dimension. The main tool is given by a detailed analysis of the Green function for the linearized problem. (C) 2007 Wiley Periodicals, Inc.