On the interior regularity of weak solutions to the 2-D incompressible Euler equations

TL;DR

This paper proves that weak solutions to the 2-D incompressible Euler equations are exponentially integrable under minimal conditions, leading to interior regularity results that are local and avoid boundary assumptions, with implications for Navier-Stokes equations.

Contribution

It introduces a new local approach to interior regularity for Euler equations, avoiding boundary conditions and gradient estimates, and extends to a modified proof of Serrin's regularity criterion for Navier-Stokes.

Findings

Weak solutions are exponentially integrable under minimal conditions.

The method is local and avoids boundary and gradient estimates.

The approach provides an alternative proof of Serrin's criterion for Navier-Stokes.

Abstract

We consider the 2-D incompressible Euler equations in a bounded domain and show that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity result $$u \in L_{\rm loc}^{2+\varepsilon}(\Omega_T) \implies {\rm regularity}$$ for weak solutions in the energy space $L_t^\infty L_x^2$ satisfying appropriate vorticity estimates. The argument is completely local in nature as the result follows from the structural properties of the equation alone, while completely avoiding all sorts of boundary conditions and related gradient estimates. To the best of our knowledge, the approach we follow is new in the context of Euler equations and provides an alternative look at interior regularity issues. We also show how our method can be used to give a modified proof of the classical Serrin condition for the regularity of the Navier-Stokes equations in any dimension.