Weakly Nonlinear-Dissipative Approximations of Hyperbolic-Parabolic Systems with Entropy

TL;DR

This paper develops weakly nonlinear-dissipative approximations for hyperbolic-parabolic systems with entropy, establishing conditions for global solutions and uniqueness, and applies the theory to the compressible Navier-Stokes system.

Contribution

It introduces a framework for weakly nonlinear-dissipative approximations with entropy, proving existence, uniqueness, and strict dissipation criteria, and applies it to fluid dynamics models.

Findings

Global weak solutions exist for strictly dissipative approximations.

Weak-strong uniqueness holds for these approximations.

A Kawashima type criterion ensures strict dissipation.

Abstract

Hyperbolic-parabolic systems have spatially homogenous stationary states. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern perturbations of these constant states. These approximations are quadratically nonlinear. When the original system has an entropy, the approximation is formally dissipative in a natural Hilbert space. We show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that Hilbert space. We also prove a weak-strong uniqueness theorem for it. In addition, we give a Kawashima type criterion for this approximation to be strictly dissipative. We apply the theory to the compressible Navier-Stokes system.