Weak-strong uniqueness and vanishing viscosity for incompressible Euler equations in exponential spaces

TL;DR

This paper establishes weak-strong uniqueness and vanishing viscosity limits for incompressible Euler equations with solutions in exponential integrability spaces, extending the understanding of solution behavior under less regular conditions.

Contribution

It proves weak-strong uniqueness for Euler equations assuming exponential integrability of the symmetric gradient and demonstrates convergence of Navier-Stokes solutions to Euler solutions under these conditions.

Findings

Weak-strong uniqueness holds in exponential spaces.

Vanishing viscosity solutions converge to Euler solutions.

Results extend solution regularity frameworks for incompressible flows.

Abstract

In the class of admissible weak solutions, we prove a weak-strong uniqueness result for the incompressible Euler equations assuming that the symmetric part of the gradient belongs to $L^1_{\rm loc}([0,+\infty);L^{\rm exp}(\mathbb{R}^d;\mathbb{R}^{d\times d}))$, where $L^{\rm exp}$ denotes the Orlicz space of exponentially integrable functions. Moreover, under the same assumptions on the limit solution to the Euler system, we obtain the convergence of vanishing-viscosity Leray--Hopf weak solutions of the Navier--Stokes equations.