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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Veröffentlicht in:	Archive for rational mechanics and analysis Jg. 206; H. 2; S. 515 - 616
Hauptverfasser:	Coutand, Daniel, Shkoller, Steve
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer-Verlag 01.11.2012 Springer Nature B.V
Schlagworte:	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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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.
AbstractList	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. 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 ^sub γ^ ρ ^sup 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.[PUBLICATION ABSTRACT]
Author	Shkoller, Steve Coutand, Daniel
Author_xml	– sequence: 1 givenname: Daniel surname: Coutand fullname: Coutand, Daniel email: d.coutand@ma.hw.ac.uk organization: CANPDE, Maxwell Institute for Mathematical Sciences, Department of Mathematics, Heriot-Watt University – sequence: 2 givenname: Steve surname: Shkoller fullname: Shkoller, Steve organization: Department of Mathematics, University of California
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Keywords	Compressible Euler Equation Sobolev Embedding Theorem Physical Vacuum Euler Equation Free Boundary Problem
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Snippet	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 )... 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) =...
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SubjectTerms	Classical Mechanics Complex Systems Eulers equations Fluid- and Aerodynamics Mathematical and Computational Physics Physics Physics and Astronomy Theoretical
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Title	Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum
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