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

TL;DR

This paper analyzes the long-term behavior of smooth solutions to entropy dissipative hyperbolic systems, showing they approach equilibrium at specific rates and can be approximated by linearized or parabolic models.

Contribution

It provides a detailed analysis of the asymptotic decay rates and approximation methods for solutions of dissipative hyperbolic systems under the Shizuta-Kawashima condition.

Findings

Solutions approach equilibrium in Lp-norm at rate O(t^(-m/2(1-1/p)))

Approximation of the conservative part by linearized hyperbolic operator with faster convergence

Use of Green function analysis for the linearized problem

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 Lp-norm at a rate O(t^(-m/2(1-1/p))), as t tends to $\infty$, for p in [min (m,2),+ \infty]. Moreover, we can show that we can approximate, with a faster order of convergence, theconservative 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. The main tool is given by a detailed analysis of the Green function for the linearized problem.