A BKM-type criterion for the Euler equations
Mustafa Sencer Aydın
TL;DR
The paper develops new Beale–Kato–Majda–type blow-up criteria for the 3D incompressible Euler equations in various domains, proving that smoothness can persist under a finite $L^2_t$-in-time control of first-order tangential velocity derivatives, i.e. $\int_0^T \|u(s)\|_{W^{1,\infty}_\text{tan}}^2 ds < \infty$. It also establishes a mixed criterion combining domain-decomposed conormal-derivative control of the velocity with vorticity bounds, and a criterion tailored to Sobolev conormal spaces (AK1). The results extend to curved boundaries via a local chart/partition-of-unity approach, and to conormal-solution frameworks, with rigorous proofs of the flat-boundary cases and curvature-robust extensions. A key technical theme is the vortex-stretching term $\omega\cdot\nabla u$, shown to be governed by tangential and conormal derivatives, enabling continuation criteria that go beyond the classical $L^1_t L^\infty$ vorticity norm. The work also provides a conormal-space well-posedness result, connecting geometric regularity with blow-up diagnostics in fluid dynamics.
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 $L^1_t L^\infty(Ω_1)$ norm of the vorticity and the $L^2_t L^\infty(Ω_2)$ norm of the first order conormal derivatives of the velocity where $Ω_1 \cup Ω_2 = Ω$ 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}.