A priori estimates for the free-boundary Euler equations with surface tension in three dimensions

Abstract

We derive a priori estimates for the incompressible free-boundary Euler equations with surface tension in three spatial dimensions. Working in Lagrangian coordinates, we provide a priori estimates for the local existence when the initial velocity, which is rotational, belongs to $H^3$ and the trace of initial velocity on the free boundary to $H^{3.5}$, thus lowering the requirement on the regularity of initial data in the Lagrangian setting. Our methods are direct and involve three key elements: estimates for the pressure, the boundary regularity provided by the mean curvature, and the Cauchy invariance.

References in corpus (2)

• Gravity capillary standing water waves
• Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces

Cited by in corpus (7)

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• Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics
• On the Incompressible Limit for the Compressible Free-Boundary Euler Equations with Surface Tension in the Case of a Liquid
• A priori Estimates for the Incompressible Free-Boundary Magnetohydrodynamics Equations with Surface Tension
• Small scale creation for 2D free boundary Euler equations with surface tension