Localized Blow-Up Criterion for C1,α Solutions to the 3D Incompressible Euler Equations

초록

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∈ C([0 , T) ; C1,α(Ω)) ∩ L∞(0 , T; L2(Ω)) be a solution to the Euler equations in a domain Ω ⊂ R3 . If there exists a ball B⊂ Ω such that ∫0T‖ω(s)‖BMO(B)ds<+∞, where ω= ∇ × v stands for the vorticity, then v∈ C([0 , T] ; C1,α(K)) for every compact subset K⊂ 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. © 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

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