Viscosity and relaxation approximation for hyperbolic systems of conservation laws

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.