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

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 $H^3\times H^4$ regularity level of the Shatah--Zeng local well-posedness theory and imposes \textit{no} assumptions of symmetry, periodicity, graph structure, or simple connectedness.

If the maximal existence time $T<\infty$, 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 $H^{\frac{3}{2}}$, or loss of boundary regularity in $H^{2+\varepsilon}$ for any sufficiently small fixed $\varepsilon>0$; (iii) loss of $H^{\frac{5}{2}}$ regularity of the normal boundary velocity; or (iv) $L^1_tL^\infty$ blow-up of the interior velocity gradient.

For simply connected domains, the interior alternative admits a refinement involving only the $L^1_tL^\infty$-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.