A Generalized Beale-Kato-Majda Breakdown Criterion for the free-boundary problem in Euler Equations with Surface Tension

It is shown in Ferrari \cite{Ferrari-1993CMP} that if $[0, T^*)$ is the maximal time interval of existence of a smooth solution of the incompressible Euler equations in a bounded, simply-connected domain in $\mathbb{R}^3$, then $\int_0^{T^*}\|\omega(t,\cdot)\|_{L^\infty}dt=+\infty$, where $\omega$ is the vorticity of the flow. Ferrari's result generalizes the classical Beale-Kato-Majda \cite{BKM-1984CMP}'s breakdown criterion in the case of a bounded fluid domain. In this manuscript, we show a breakdown criterion for a smooth solution of the Euler equations describing the motion of an incompressible fluid in a bounded domain in $\mathbb{R}^3$ with a free surface boundary. The fluid is under the influence of surface tension. In addition, we show that our breakdown criterion reduces to the one proved by Ferrari \cite{Ferrari-1993CMP} when the free surface boundary is fixed. Specifically, the additional control norms on the moving boundary will either become trivial or stop showing up if the kinematic boundary condition on the moving boundary reduces to the slip boundary condition.