A BKM-type criterion for the Euler equations

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 $L^{\mathrm{\infty }}$ 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 $L_t^1{L}^{\mathrm{\infty }}({\mathrm{\Omega }}_1)$ norm of the vorticity and the $L_t^2{L}^{\mathrm{\infty }}({\mathrm{\Omega }}_2)$ norm of the first order conormal derivatives of the velocity where ${\mathrm{\Omega }}_1\cup {\mathrm{\Omega }}_2=\mathrm{\Omega }$ 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}.