A BKM-type criterion for the Euler equations

TL;DR

This paper introduces a new blow-up criterion for the incompressible Euler equations, involving Sobolev spaces and tangential derivatives, applicable to various domains and including vorticity-based conditions.

Contribution

It establishes a novel BKM-type blow-up criterion involving tangential derivatives and vorticity norms, extending applicability to diverse domains and Sobolev conormal spaces.

Findings

New blow-up criterion involving $L^2_t$ of $L^ abla_ ext{infty}$ norms.

Applicability to multiple domain types including curved boundaries.

Inclusion of vorticity and conormal derivatives in blow-up conditions.

Abstract

We establish a new BKM-type blow-up criterion for solutions of the incompressible Euler equations that belong to Sobolev or H\" older spaces. Our criterion involves the $L^2$ norm in time of the $L^\infty$ norm of the first order tangential derivatives. Moreover, it applies to various domains such as the full space, the half-space, torus, (in)finite channel, and domains with curved boundaries. Additionally, we provide a mixed criterion involving the $L^1_t L^\infty(\Omega_1)$ norm of the vorticity and the $L^2_t L^\infty(\Omega_2)$ norm of the first order conormal derivatives of the velocity where $\Omega_1 \cup \Omega_2 = \Omega$ is a suitable decomposition of the physical space. Finally, we prove a blow-up criterion for the class of solutions that belong to the Sobolev conormal spaces that is recently constructed in~\cite{AK1}.