Local Well-Posedness for the Motion of a Compressible, Self-Gravitating Liquid with Free Surface Boundary

Abstract

We establish the local well-posedness for the free boundary problem for the compressible Euler equations describing the motion of liquid under the influence of Newtonian self-gravity. We do this by solving a tangentially-smoothed version of Euler's equations in Lagrangian coordinates which satisfies uniform energy estimates as the smoothing parameter goes to zero. The main technical tools are delicate energy estimates and optimal elliptic estimates in terms of boundary regularity, for the Dirichlet problem and Green's function.

On the Motion of a Compressible Gravity Water Wave with Vorticity

“…Proof. The proof is largely similar to what is in [19,Appendix B] and so we shall only sketch the details. The main idea here is to apply the div-curl estimate on ∇ã f 2Ḣ r .…”

“…Also, for its application in the compressible free-boundary Euler equations modeling a liquid in a bounded domain, Coutand-Hole-Shkoller [7] obtained the LWP for the case with surface tension and Ginsberg-Lindblad-Luo [19] obtained the LWP for the self-gravitating liquid. However, here we use a different set of approximate problems by adapting what appears in [19] which yields a simpler construction of the sequence of approximate solutions. This will be discussed in Sect.…”

“…In Ginsberg-Lindblad-Luo [19], the compressible Euler equations are approximated by a "fully smoothed system", in the sense that all variables are replaced by their smoothed version. Specifically, they smooth the velocity vector field in the tangential direction first and then obtain the smoothed flow map by integrating it in time (see [19] Section 4). In this paper, however, we smooth the flow map directly and through this we replace the nonlinear coefficients a µα by their smoothed versioñ a µα in (1.20).…”

A priori estimates for relativistic liquid bodies

“…Existence for compressible gaseous bodies was also established using a different approach in [38]. Recently, a new approach to establishing the local-intime existence of solutions for compressible, self-gravitating, liquid bodies has been developed in [30]. For other related results in the non-relativistic setting, which includes other approaches to a priori estimates, existence on small and large time scales, and coupling to Newtonian gravity, see the works [2,3,5,8,9,13,18,24,27,28,33,32,35,36,45,46,47,48,50,57,58,66] and references cited therein.…”

“…(a) As in [40], the boundary terms g|φ ΓT and (∂ t q − p)∂ t u|φ ΓT are defined via the expressions 30 g|φ ΓT = ν(g) + ∂ Σ ν Σ g|φ ΩT + g|ν(φ) ΩT , (E. 30) and…”