On the Motion of a Compressible Gravity Water Wave with Vorticity

Abstract

We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary with unbounded initial domain. The liquid is under influence of gravity but without surface tension. Our fluid is not assumed to be irrotational. But the physical sign condition needs to be assumed on the free boundary. We generalize the method used in [20] to prove the energy estimates in an unbounded domain up to arbitrary order. In addition to that, the a priori energy estimates are in fact uniform in the sound speed κ. As a consequence, we obtain the convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of [20] to when you have an unbounded domain. On the other hand, we prove that there are initial data satisfying the compatibility condition in some weighted Sobolev spaces, and this will propagate within a short time interval, which is essential for proving long time existence for slightly compressible irrotational water waves.

Incompressible limit for the free surface Navier-Stokes system

“…In [41], Lindblad-Luo prove uniform apriori estimates for the free boundary compressible Euler equations in the case of a bounded reference domain. More recently, this result is extended by Luo [44] for unbounded reference domains and by Disconzi-Luo [20] for a bounded reference domain but with surface tension. All these results are based on the assumption that the initial datum is sufficiently well-prepared in the sense that the time derivatives up to at least order two are bounded initially, an assumption which is stronger than the usual well-prepared data assumption which requires one time derivative to be bounded initially.…”

A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid

“…Earlier and related works are [21,24,25,28,31,62,63,72,79,92,93]. Further, and more recent results, are [50,61,76,78].…”

A regularity result for the incompressible magnetohydrodynamics equations with free surface boundary

“…Important progress has been made for both incompressible and compressible flows, with or without surface tension, and with or without vorticity. Without attempting to be exhaustive, we refer [1,5,6,9,18,22,23,24,25,26,28,29,32,33,34,40,41,42] for more details.…”