Well-posedness of 3D vortex sheets with surface tension. (English) Zbl 1130.76016

Ambrose, David M.; Masmoudi, Nader

Commun. Math. Sci. 5, No. 2, 391-430 (2007).

Summary: We prove well-posedness for the initial value problem for a vortex sheet in 3D fluids, in the presence of surface tension. We first reformulate the problem by making a favorable choice of variables and parameterizations. We then perform energy estimates for the evolution equations. It is important to note that the Kelvin-Helmholtz instability is present for the vortex sheet in the absence of surface tension. Accordingly, we must construct the energy functional carefully with an eye toward the regularization of this instability. Well-posedness follows from the estimates.

MSC:

76B03	Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76B47	Vortex flows for incompressible inviscid fluids
76B45	Capillarity (surface tension) for incompressible inviscid fluids
35Q35	PDEs in connection with fluid mechanics

Keywords:

energy estimates; Kelvin-Helmholtz instability; regularization