Classification of finite-time blow-up mechanisms for the incompressible free-boundary Euler equations with surface tension

2025/07/14 by Chengchun Hao, Tao Luo, Hao, Chengchun +3
#math.AP

paper · pdf · doi:10.48550/arxiv.2507.10032

Abstract

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 H^2+ε for any sufficiently small fixed ε>0; (iii) loss of H^(5)/(2) regularity of the normal boundary velocity; or (iv) L¹_tL^∞ blow-up of the interior velocity gradient. For simply connected domains, the interior alternative admits a refinement involving only the L¹_tL^∞-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.

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