On a class of generalized solutions to equations describing incompressible viscous fluids

TL;DR

This paper introduces dissipative solutions for viscous fluid equations with monotone stress dependence, proving existence, uniqueness, and conditions under which solutions match classical ones.

Contribution

It defines a new class of solutions for viscous fluids, extending measure-valued concepts, and establishes their fundamental properties including weak-strong uniqueness.

Findings

Existence of dissipative solutions is proven.

Dissipative solutions satisfy weak-strong uniqueness.

Regular dissipative solutions coincide with classical solutions.

Abstract

We consider a class of viscous fluids with a general monotone dependence of the viscous stress on the symmetric velocity gradient. We introduce the concept of dissipative solution to the associated initial boundary value problem inspired by the measure-valued solutions for the inviscid (Euler) system. We show the existence as well as the weak-strong uniqueness property in the class of dissipative solutions. Finally, the dissipative solution enjoying certain extra regularity coincides with a strong solution of the same problem.