Relative energy inequality and weak-strong uniqueness]{Relative energy inequality and weak-strong uniqueness for an isothermal non-Newtonian compressible fluid

TL;DR

This paper derives a relative energy inequality for weak solutions of 3D non-Newtonian compressible Navier-Stokes equations and proves weak-strong uniqueness, ensuring weak solutions match strong solutions when both exist.

Contribution

It introduces a relative energy inequality for non-Newtonian compressible fluids and establishes weak-strong uniqueness results for these systems.

Findings

Relative energy inequality derived for weak solutions.

Standard energy inequality implies the relative energy inequality.

Weak and strong solutions coincide under certain conditions.

Abstract

Our paper deals with three-dimensional nonsteady Navier-Stokes equations for non-Newtonian compressible fluids. It contains a~derivation of the relative energy inequality for the weak solutions to these equations. We show that the standard energy inequality implies the relative energy inequality. Consequently, the relative energy inequality allows us to achieve a weak-strong uniqueness result. In other words, we present that the weak solution of the Navier-Stokes system coincides with the strong solution emanated from the same initial conditions as long as the weak solution exists.