Energy equality of the weak solutions to non-Newtonian fluids equations

TL;DR

This paper establishes new conditions under which weak solutions to 3D non-Newtonian fluid equations satisfy energy equality, advancing understanding of solution uniqueness.

Contribution

It introduces Sobolev multiplier space criteria that ensure energy equality for weak solutions, linking to the uniqueness problem in non-Newtonian fluid dynamics.

Findings

Derived sufficient conditions for energy equality using Sobolev multiplier spaces

Connected energy equality to the uniqueness of weak solutions in non-Newtonian fluids

Provided insights analogous to Onsager's conjecture for non-Newtonian fluids

Abstract

In this paper, we study the problem of energy equality for weak solutions of the 3D incompressible non-Newtonian fluid equations with initial value conditions. We derive new sufficient conditions via Sobolev multiplier spaces that guarantee the validity of the energy equality. Moreover, the aforementioned equations are often associated with the uniqueness problem of weak solutions for non-Newtonian fluids, which, in a certain sense, constitutes the positive counterpart of Onsager's conclusion for non-Newtonian fluids.