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Well-posedness of electrohydrodynamic interfacial waves under tangential electric field. (English) Zbl 1464.35259

Summary: We consider the motion of the interface between two inviscid, incompressible, and dielectric fluids with different densities and permittivities, in the presence of a uniform electric field acting in a direction parallel to the undisturbed configuration. The system is assumed to be irrotational except the interface where the discontinuity of the tangential velocity induces vorticity. In this paper, we establish the local existence and uniqueness theory for the initial-value problem in Sobolev spaces for interfacial electrohydrodynamics. As we show, this system is locally well-posed in both two and three dimensions when surface tension is taken into account. More importantly, the tangential electric field provides a significant stabilizing effect for the two-dimensional problem (with a one-dimensional interface) such that we can prove the local-in-time well-posedness for small data even if one neglects the surface tension.

MSC:

35Q35 PDEs in connection with fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
76T06 Liquid-liquid two component flows
76D45 Capillarity (surface tension) for incompressible viscous fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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