Singular limits for the Riemann problem. General diffusion, relaxation, and boundary conditions

TL;DR

This paper studies the behavior of solutions to nonlinear hyperbolic systems with diffusion and relaxation effects, showing convergence to Riemann problem solutions as diffusion vanishes, applicable to boundary-value problems.

Contribution

It establishes the existence of smooth, self-similar solutions with bounded variation for systems with general diffusion matrices, without requiring genuine nonlinearity.

Findings

Solutions converge to Riemann problem solutions as diffusion tends to zero

Existence of smooth, self-similar solutions with bounded total variation

Results extend to relaxation and boundary-value problems

Abstract

We consider self-similar approximations of nonlinear hyperbolic systems in one space dimension with Riemann initial data and general diffusion matrix. We assume that the matrix of the system is strictly hyperbolic and the diffusion matrix is close to the identity. No genuine nonlinearity assumption is required. We show the existence of a smooth, self-similar solution which has bounded total variation, uniformly in the diffusion parameter. In the zero-diffusion limit, the solutions converge to a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation and the boundary-value problem in a half-space for the same regularizations.