Vanishing viscosity limit for hyperbolic system of Temple class in 1-d with nonlinear viscosity

TL;DR

This paper proves the global existence of smooth solutions and the vanishing viscosity limit for a 1D hyperbolic system of Temple class with nonlinear viscosity, establishing convergence to the hyperbolic system's solution as viscosity approaches zero.

Contribution

It demonstrates the global existence of solutions and the zero viscosity limit for a specific class of hyperbolic systems with nonlinear viscosity, extending previous results.

Findings

Solutions exist globally for small total variation initial data.

Solutions to the parabolic system converge to the hyperbolic system's solution as viscosity vanishes.

The zero diffusion limit matches previous established results.

Abstract

We consider hyperbolic system with nonlinear viscosity such that the viscosity matrix $B(u)$ is commutating with $A(u)$ the matrix associated to the convective term. The drift matrix is assumed to be Temple class. First, we prove the global existence of smooth solutions for initial data with small total variation. We show that the solution to the parabolic equation converges to a semi-group solution of the hyperbolic system as viscosity goes to zero. Furthermore, we prove that the zero diffusion limit coincides with the one obtained in [Bianchini and Bressan, Indiana Univ. Math. J. 2000].