Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum

Abstract

We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = C γ ρ γ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss-Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy's inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.

Global stability of steady states in the classical Stefan problem for general boundary shapes

“…Because the liquid phase Ω(t) is characterized by the set {x ∈ R d : p(x, t) > 0}, we shall consider initial data p 0 > 0 in Ω. Thanks to (1.1a), the parabolic Hopf lemma implies that ∂ n p(t) < 0 on Γ (t) for t > 0, so we impose the non-degeneracy condition (also known as the Rayleigh-Taylor sign condition in fluid mechanics [1][2][3][4][5][6]):…”

Short-Time Structural Stability of Compressible Vortex Sheets with Surface Tension

“…Also, the case of a gas-vacuum boundary has been solved under certain boundary conditions by Coutand and Shkoller [13] and Jang and Masmoudi [16].…”

“…Rather than follow the Nash-Moser scheme of [4,9,17,20], which can often be very technical, we introduce a degenerate artificial viscosity type regularisation inspired by [10,13,16], and prove a priori estimates on the nonlinear equations in a somewhat similar manner to [6], but using a change of coordinates which is a simple lift of the graph of the free-surface, similar to that of [7], rather than Lagrangian coordinates.…”

“…We call the operator ∂ j 0 the initial time derivative operator of order j associated to the system (9)- (13).…”

“…be initial data for the system (9)- (13). We say that the initial data ( p 0 ± , u 0 ± , s 0 ± , f 0 ) satisfy the compatibility conditions for the system (9)-(13) up to order k, for integer k 0, if the following holds for all x ∈ R 2 :…”