We consider the dynamics of a layer of an incompressible electrically conducting fluid interacting with the magnetic field in a two-dimensional horizontally periodic setting. The upper boundary is in contact with the atmosphere, and the lower boundary is a rigid flat bottom. We prove the global well-posedness of the inviscid and resistive problem with surface tension around a non-horizontal uniform magnetic field; moreover, the solution decays to the equilibrium almost exponentially. One of the key observations here is an induced damping structure for the fluid vorticity due to the resistivity and transversal magnetic field.
For the free-boundary incompressible Euler equations, the early works were focused on the irrotational fluids, which began with the local well-posedness for the small initial data @cite_22 @cite_38 @cite_1 and was generalized to the general initial data by the breakthrough of Wu @cite_33 @cite_39 , see also Lannes @cite_0 and Ambrose and Masmoudi @cite_13 @cite_26 . For the general incompressible Euler equations without the irrotational assumption, the first local well-posedness was obtained by Lindblad @cite_5 for the case without surface tension and by Coutand and Shkoller @cite_2 for the case with (and without) surface tension, see also Christodoulou and Lindblad @cite_43 , Schweizer @cite_24 , Shatah and Zeng @cite_14 and Zhang and Zhang @cite_12 . We also refer to the local well-posedness in conormal Sobolev spaces as the inviscid limits of the free-boundary incompressible Navier-Stokes equations, see Masmoudi and Rousset @cite_8 and the authors @cite_11 .
@cite_38 @cite_26 @cite_14 @cite_22 @cite_33 @cite_8 @cite_1 @cite_39 @cite_0 @cite_43 @cite_24 @cite_2 @cite_5 @cite_13 @cite_12 @cite_11
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We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order In this paper, we consider in three dimensions the motion of a general inviscid, incompressible fluid with a free interface that separates the fluid region from the vacuum. We assume that the fluid region is below the vacuum and that there is no surface tension on the free surface. Then we prove the local well-posedness of the free boundary problem in Sobolev space provided that there is no self-intersection point on the initial surface and under the stability assumption that with ξ being restricted to the initial surface. © 2007 Wiley Periodicals, Inc. We study the inviscid limit of the free boundary Navier–Stokes equations. We prove the existence of solutions on a uniform time interval by using a suitable functional framework based on Sobolev conormal spaces. This allows us to use a strong compactness argument to justify the inviscid limit. Our approach does not rely on the justification of asymptotic expansions. In particular, we get a new existence result for the Euler equations with free surface from the one for Navier–Stokes. Abstract 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.