A Lagrangian Interior Regularity Result for the Incompressible Free Boundary Euler Equation with Surface Tension

Abstract

We consider the three-dimensional incompressible free-boundary Euler equations in a bounded domain and with surface tension. Using Lagrangian coordinates, we establish a priori estimates for solutions with minimal regularity assumptions on the initial data.

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• Gravity capillary standing water waves
• On the Hamiltonian for water waves
• A priori estimates for the motion of a self-gravitating incompressible liquid with free surface boundary
• Local well-posedness and break-down criterion of the incompressible Euler equations with free boundary