Well-posedness of the Free-surface Incompressible Euler Equations with or without Surface Tension
نویسندگان
چکیده
We develop a new methodology for treating free boundary problems in mechanics, and use it to prove local-in-time well-posedness in Sobolev spaces for the freesurface 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.
منابع مشابه
Well-Posedness of the Free-Boundary Compressible 3-D Euler Equations with Surface Tension and the Zero Surface Tension Limit
We prove that the 3-D compressible Euler equations with surface tension along the moving free-boundary are well-posed. Specifically, we consider isentropic dynamics and consider an equation of state, modeling a liquid, given by Courant and Friedrichs [8] as p(ρ) = αρ − β for consants γ > 1 and α, β > 0. The analysis is made difficult by two competing nonlinearities associated with the potential...
متن کاملLocal Well-posedness of the Viscous Surface Wave Problem without Surface Tension
We consider a viscous fluid of finite depth below the air, occupying a threedimensional domain bounded below by a fixed solid boundary and above by a free moving boundary. The domain is allowed to have a horizontal cross-section that is either periodic or infinite in extent. The fluid dynamics are governed by the gravity-driven incompressible Navier-Stokes equations, and the effect of surface t...
متن کاملA Simple Proof of Well-posedness for the Free-surface Incompressible Euler Equations
The purpose of this this paper is to present a new simple proof for the construction of unique solutions to the moving free-boundary incompressible 3-D Euler equations in vacuum. Our method relies on the Lagrangian representation of the fluid, and the anisotropic smoothing operation that we call horizontal convolution-by-layers. The method is general and can be applied to a number of other movi...
متن کاملThe Viscous Surface-Internal Wave Problem: Global Well-Posedness and Decay
We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a threedimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state a...
متن کاملA Priori Estimates for Free Boundary Problem of Incompressible Inviscid Magnetohydrodynamic Flows
In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid MHD equations in all physical spatial dimensions n = 2 and 3 by adopting a geometrical point of view used in [4], and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the out...
متن کامل