Well‐posedness in smooth function spaces for moving‐boundary 1‐D compressible euler equations in physical vacuum

Abstract

The free-boundary compressible one-dimensional Euler equations with moving physical vacuum boundary are a system of hyperbolic conservation laws that are both characteristic and degenerate. The physical vacuum singularity (or rate of degeneracy) requires the sound speed c 2 D 1 to scale as the square root of the distance to the vacuum boundary and has attracted a great deal of attention in recent years. We 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. The proof is founded on a new higher-order, Hardy-type inequality in conjunction with an approximation of the Euler equations consisting of a particular degenerate parabolic regularization. Our regular solutions can be viewed as degenerate viscosity solutions.

Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping

“…[5,11]) and continuation arguments. The uniqueness of the smooth solutions can be obtained as in section 11 of [20].…”

Nonlinear Asymptotic Stability of the Lane-Emden Solutions for the Viscous Gaseous Star Problem with Degenerate Density Dependent Viscosities

“…When the viscosities are positive constants, the global-in-time spherically symmetric solution to the free boundary problem (1.1) and its nonlinear asymptotic stability toward the Lane-Emden solution were proved in [30] for 4/3 < γ < 2 (the stable index), by establishing the global-in-time regularity uniformly up to the vacuum boundary of solutions capturing an interesting behavior called the physical vacuum which states that the sound speed c = p ′ (ρ) is C 1/2 -Hölder continuous near the vacuum boundary (cf. [2,3,13,16,23,25,42]), as long as the initial datum is a suitably small perturbation of the Lane-Emden solution with the same total mass. The large time asymptotic convergence of the global strong solution, in particular, the convergence of the vacuum boundary and the uniform convergence of the density, to those of the Lane-Emden solution with detailed convergence rates as the time goes to infinity are given in [30] when the viscosities are constant.…”

“…And the initial density is supposed to satisfy the following condition: (r) ∼ R 0 − r as r close to R 0 , (1.5) that is, the initial sound speed is C 1/2 -Hölder continuous across the vacuum boundary, which is called the physical vacuum for the compressible inviscid flows (cf. [2,3,13,25,42]). …”

Global Stability and Decay for the Classical Stefan Problem

“…Let (q 0 , h 0 ) satisfy the Taylor sign condition (18), the strict positivity assumption (17), and the compatibility conditions (19), (20). Let K be defined as in (21).…”

“…Under the above assumptions, we proved in Hadžić & Shkoller [29] that (1) is indeed well-posed. 1 This type of stability condition dates back to the early work of Lord Rayleigh [46] and Taylor [48] in fluid mechanics, and appears as a necessary well-posedness condition on the initial data in many free-boundary problems wherein the effects of surface tension are ignored; examples include the Hele-Shaw cell, the water waves equations [50], and the full Euler equations in both incompressible [15] and compressible form [18,17].…”