Existence of vanishing physical viscosity solutions of characteristic initial-boundary value problems for systems of conservation laws

TL;DR

This paper proves the existence of solutions for boundary characteristic initial-boundary value problems in conservation laws, focusing on the zero-viscosity limit and introducing new wave front-tracking techniques and boundary layer analysis.

Contribution

It establishes the existence of admissible solutions in the boundary characteristic case using a novel wave front-tracking algorithm and boundary layer estimates, applicable to physical hyperbolic-parabolic systems.

Findings

Existence of solutions in boundary characteristic cases.

Development of a new wave front-tracking algorithm.

Analysis applicable to Navier-Stokes and MHD equations.

Abstract

We consider initial boundary-value problems for nonlinear systems of conservation laws in one space variable. It is known that in general different viscous mechanisms yield different solutions in the zero-viscosity limit. Here we focus on the most technically demanding case, known as boundary characteristic case, which occurs when one of the characteristic velocities of the system vanishes. We work in small total variation regimes and assume that every characteristic field is either genuinely nonlinear or linearly degenerate. We establish existence of admissible solutions satisfying a boundary condition consistent with the vanishing viscosity approximation given by a large class of physical (that is, mixed hyperbolic-parabolic) systems. In particular, our results apply to the zero-viscosity limit of the Navier-Stokes and viscous MHD equations, written in both Eulerian and Lagrangian coordinates. Our analysis relies on a fine boundary layers analysis and is based on the introduction of a new wave front-tracking algorithm. From the technical viewpoint, the most innovative elements are i) a new class of interaction estimates for boundary layers and boundary characteristic wave fronts hitting the boundary, which yields the introduction of a new Glimm-type functional; ii) a detailed analysis of the behavior of the wave front-tracking algorithm close to the boundary, which in turn yields relevant information on the limit.