2025/07/14 by Chengchun Hao, Tao Luo, T. Luo +5
Earth and Planetary Sciences · Engineering · Mathematics · #Aquatic and Environmental Studies #Boundary (topology) #Bounded function #Compressibility #Computational Fluid Dynamics and Aerodynamics #Conservative vector field #Curvature #Domain (mathematical analysis) #Euler equations #Free surface #Navier-Stokes equation solutions #Vector field #math.AP
paper · pdf · doi:10.48550/arxiv.2507.10032
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We establish a blow-up criterion for strong solutions of the three-dimensional incompressible Euler equations with surface tension in a bounded domain with a closed moving free boundary. The criterion is formulated at the H3× H4 regularity level of the Shatah--Zeng local well-posedness theory and imposes no assumptions of symmetry, periodicity, graph structure, or simple connectedness. If the maximal existence time T<∞, then at least one of the following four mechanisms must occur: (i) first self-intersection of the free boundary; (ii) loss of mean curvature regularity in H(3)/(2), or loss of boundary regularity in H2+ε for any sufficiently small fixed ε>0; (iii) loss of H(5)/(2) regularity of the normal boundary velocity; or (iv) L1tL^∞ blow-up of the interior velocity gradient. For simply connected domains, the interior alternative admits a refinement involving only the L1tL^∞-norm of the vorticity, and this refinement recovers exactly the classical Beale--Kato--Majda criterion in the fixed-boundary case. For irrotational flows in the simply connected free-boundary setting, the criterion reduces to the three boundary mechanisms.