## Large-Time Behavior of Entropy Solutions of
Conservation Laws

#### Authors: Gui-Qiang Chen and Hermano Frid

#### Title: Large-Time Behavior of Entropy Solutions of
Conservation Laws

**Abstract**

We are concerned with the large-time behavior of discontinuous entropy
solutions for hyperbolic systems of conservation laws.
We present two analytical approaches and explore their applications
to the asymptotic problems for discontinuous entropy solutions.
These approaches allow the solutions of arbitrarily large oscillation
without apriori assumption on the ways from which the solutions come.
The relation between the large-time behavior of entropy solutions
and the uniqueness of Riemann solutions leads to an extensive study
of the uniqueness problem.
We use a direct method to show the large-time behavior of large $L^\infty$
solutions for a class of $m\times m$ systems including a model in
multicomponent chromatography;
we employ the uniqueness of Riemann solutions and the convergence of
self-similar scaling sequence of solutions to show the asymptotic behavior
of large $BV$ solutions for the $3\times 3$ system of Euler equations in
thermoelasticity.
These results indicate that the Riemann solution is the unique attractor
of large discontinuous entropy solutions, whose initial data
are $L^\infty\cap L^1$ or $BV\cap L^1$ perturbation of the Riemann data,
for these systems. These approaches also work for proving the large-time
behavior of approximate solutions to hyperbolic conservation laws.

This article has appeared in:

*Journal of Differential Equations*, **152**, pages 308-357 (1999)

This paper is available in the following formats:

A closely related paper is
Decay of Entropy Solutions of Nonlinear Conservation Laws.

Another closely related paper is
Decay of Entropy Solutions in $L^\infty$ for Multidimensional
Conservation Laws.

**Author Address**
Gui-Qiang Chen
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
gqchen@math.nwu.edu