VISCOSITY AND RELAXATION APPROXIMATIONS

FOR HYPERBOLIC SYSTEMS OF CONSERVATION LAWS

Athanasios E. Tzavaras

Abstract. These lecture notes deal with the approximation of conservation laws via viscosity or relaxation. The following topics are covered:

The general structure of viscosity and relaxation approximations is discussed, as suggested by the second law of thermodynamics, in its form of the Clausius-Duhem inequality. This is done by reviewing models of one dimensional thermoviscoelastic materials, for the case of viscous approximations, and thermomechanical theories with internal variables, for the case of relaxation.

The method of self-similar zero viscosity limits is an approach for constructing solutions to the Riemann problem, as zero-viscosity limits of an elliptic regularization of the Riemann operator. We present recent results on obtaining uniform BV estimates, in a context of strictly hyperbolic systems for Riemann data that are sufficiently close. The structure of the emerging solution, and the connection with shock admissibility criteria is discussed.

The problem of constructing entropy weak solutions for hyperbolic conservation laws via relaxation approximations is considered. We discuss compactness and convergence issues for relaxation approximations converging to the scalar conservation law, in a BV framework, and to the equations of isothermal elastodynamics, via compensated compactness.

Contents

1. Introduction

2. The Structure induced by Continuum Thermomechanics a. Thermomechanical theories in one-space dimension b. The constitutive theory of thermoviscoelasticity c. A hierarchy of thermomechanical theories d. Materials with internal variables e. A thermomechanical model with stress relaxation

3. Zero-Viscosity Limits for the Scalar Conservation Law

4. The Riemann Problem and Self-Similar Viscosity Limits a. The problem of self-similar viscosity limits b. The connection with shock profiles c. The scalar conservation law d. BV stability for self-similar viscosity limits e. The relation with the problem of viscosity limits

5. Relaxation Approximations of Hyperbolic Conservation Laws a. The structure of relaxation approximations b. The scalar multi-d conservation law via relaxation c. A relaxation limit to the equations of isothermal elastodynamics

References

Department of Mathematics, University of Wisconsin, Madison, WI 53706, and Institute of Applied and Computational Mathematics, FORTH, 711 10 Heraklion, Crete. E-mail: tzavaras@math.wisc.edu

Research partially supported by the Office of Naval Research, the National Science Foundation, and the TMR programme HCL # ERBFMRXCT960033.

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