Vanishing viscosity limit for $n\times n$ hyperbolic system of conservation laws in 1-d with nonlinear viscosity: Part-I Uniform BV estimates

TL;DR

This paper establishes uniform BV bounds over time for solutions to a 1-D hyperbolic conservation law system with nonlinear viscosity, under small initial data and commutation conditions, as the viscosity parameter approaches zero.

Contribution

It provides the first uniform BV estimates for nonlinear viscous approximations of hyperbolic systems with commuting matrices, advancing understanding of the vanishing viscosity limit.

Findings

Proves global uniform BV bounds for solutions with small initial data.

Demonstrates the importance of matrix commutation in obtaining estimates.

Supports the convergence analysis of viscous approximations to hyperbolic systems.

Abstract

We consider the following parabolic approximation for hyperbolic system of conservation laws in 1-D with non-singular viscosity matrix $B(u)$ and $A(u)$ strictly hyperbolic, \[u_t+A(u)u_x = \varepsilon(B(u)u_x)_x.\] We prove global in time uniform $BV$ bound for solution to this parabolic system when $\varepsilon>0$ provided that the initial data is small in $BV$ and the matrix $A(u)$ and $B(u)$ commutate.