Localized Blow-Up Criterion for $C^{ 1, \alpha }$ Solutions to the 3D Incompressible Euler Equations

Abstract

We prove a localized Beale–Kato–Majda type blow-up criterion for the 3D incompressible Euler equations in the Hölder space setting. More specifically, let $v\in C([0, T); C^{ 1, \alpha } (\Omega ))\cap L^\infty (0, T; L^2(\Omega ))$ be a solution to the Euler equations in a domain $\Omega \subset {\mathbb {R}}^3$. If there exists a ball $B\subset \Omega$ such that $\int \limits \nolimits _{0}^T \Vert \omega (s)\Vert _{ BMO(B )} ds < +\infty ,$ where $\omega = \nabla \times v$ stands for the vorticity, then $v\in C([0, T]; C^{ 1, \alpha } (K))$ for every compact subset $K \subset B$. In the proof of this result, in order to handle the time evolution of the local Hölder norm of the vorticity we use the well-known Campanato space representation for the the Hölder space, and our argument relies on the Campanato space estimates for the solution to the corresponding transport equation.