We study an incompressible ideal fluid with a free surface that is subject to surface tension; it is not assumed that the fluid is irrotational. We derive a priori estimates for smooth solutions and prove a short-time existence result. The bounds are obtained by combining estimates of energy type with estimates of vorticity type and rely on a careful study of the regularity properties of the pressure function. An adequate artificial coordinate system is used instead of the standard Lagrangian coordinates. Under an assumption on the vorticity, a solution to the Euler equations is obtained as a vanishing viscosity limit of solutions to appropriate Navier–Stokes systems.
Cite this article
Ben Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 6, pp. 753–781DOI 10.1016/J.ANIHPC.2004.11.001