Generalized solutions to models of compressible viscous fluids

TL;DR

This paper introduces dissipative solutions for compressible viscous fluid models, providing a framework that ensures weak-strong uniqueness and compatibility with classical solutions, addressing longstanding existence issues.

Contribution

It develops a novel approach based on dissipative solutions for complex fluid models with non-linear viscosity and boundary conditions, advancing the theoretical understanding.

Findings

Dissipative solutions coincide with strong solutions when they exist.

The framework ensures weak-strong uniqueness for the models.

Addresses existence of solutions in complex compressible fluid models.

Abstract

We propose a new approach to models of general compressible viscous fluids based on the concept of dissipative solutions. These are weak solutions satisfying the underlying equations modulo a defect measure. A dissipative solution coincides with the strong solution as long as the latter exists (weak-strong uniqueness) and they solve the problem in the classical sense as soon as they are smooth (compatibility). We consider general models of compressible viscous fluids with non-linear viscosity tensor and non-homogeneous boundary conditions, for which the problem of existence of global-in-time weak/strong solutions is largely open.