Weak-strong Uniqueness for Heat Conducting non-Newtonian Incompressible Fluids

TL;DR

This paper introduces a new weak solution concept for heat-conducting non-Newtonian fluids based on entropy balance, proving existence via finite element methods and establishing weak-strong uniqueness.

Contribution

It proposes a dissipative weak solution framework based on entropy, with existence proof through finite element approximations and a weak-strong uniqueness result.

Findings

First convergence result for a numerical scheme for the full system

Existence of solutions with temperature-dependent coefficients

Weak-strong uniqueness established via relative energy inequality

Abstract

In this work, we introduce a notion of dissipative weak solution for a system describing the evolution of a heat-conducting incompressible non-Newtonian fluid. This concept of solution is based on the balance of entropy instead of the balance of energy and has the advantage that it admits a weak-strong uniqueness principle, justifying the proposed formulation. We provide a proof of existence of solutions based on finite element approximations, thus obtaining the first convergence result of a numerical scheme for the full evolutionary system including temperature dependent coefficients and viscous dissipation terms. Then we proceed to prove the weak-strong uniqueness property of the system by means of a relative energy inequality.