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

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.