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

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...

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Vydáno v:Archive for rational mechanics and analysis Ročník 206; číslo 2; s. 515 - 616
Hlavní autoři: Coutand, Daniel, Shkoller, Steve
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer-Verlag 01.11.2012
Springer Nature B.V
Témata:
Classical Mechanics
Complex Systems
Eulers equations
Fluid- and Aerodynamics
Mathematical and Computational Physics
Physics
Physics and Astronomy
Theoretical
Compressible Euler Equation
Sobolev Embedding Theorem
Physical Vacuum
Euler Equation
Free Boundary Problem
ISSN:0003-9527, 1432-0673
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Shrnutí: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.
Bibliografie:SourceType-Scholarly Journals-1
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ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-012-0536-1