Well‐posedness for compressible Euler equations with physical vacuum singularity

Jang, Juhi; Masmoudi, Nader

doi:10.1002/cpa.20285

Cited by 146 publications

(153 citation statements)

References 25 publications

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“…The only existence theory for the physical vacuum singularity that we are aware of can be found in the recent paper by Jang and Masmoudi [6] for the 1D compressible gas; we refer the interested reader to the introduction in that paper for a nice history of the analysis of the 1D compressible Euler equations with damping.…”

Section: History Of Prior Results For the Compressible Euler Equationmentioning

confidence: 99%

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A Priori Estimates for the Free-Boundary 3D Compressible Euler Equations in Physical Vacuum

Coutand¹,

Lindblad

Shkoller

2010

Commun. Math. Phys.

103

130

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Abstract:We prove a priori estimates 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 vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic free-boundary system to which standard methods of symmetrizable hyperbolic equations cannot be applied.

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Section: History Of Prior Results For the Compressible Euler Equationmentioning

confidence: 99%

“…Letting ∂ 6We use Young's inequality and the fundamental theorem of calculus (with respect to t) for the last integral to find that for δ > 0,…”

Section: E(t))mentioning

confidence: 99%

A Priori Estimates for the Free-Boundary 3D Compressible Euler Equations in Physical Vacuum

Coutand¹,

Lindblad

Shkoller

2010

Commun. Math. Phys.

103

130

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“…First, using the Taylor expansion, one may verify that for γ > 4/3, 13) provided (3.1) holds for a suitably small constant ǫ 0 , where C(γ) is a positive constant depending on γ. Also,…”

Section: Lower-order Estimatesmentioning

confidence: 99%

“…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]). …”

Section: Introductionmentioning

confidence: 99%

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

Luo

Xin

Zeng

2016

Commun. Math. Phys.

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The nonlinear asymptotic stability of Lane-Emden solutions is proved in this paper for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent γ lies in the stability regime (4/3, 2), by establishing the global-in-time regularity uniformly up to the vacuum boundary for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson systems with spherical symmetry, which ensures the global existence of strong solutions capturing the precise physical behavior that the sound speed is C 1/2 -Hölder continuous across the vacuum boundary, the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of Lane-Emden solutions with detailed convergence rates, and the detailed large time behavior of solutions near the vacuum boundary. The results obtained in this paper extend those in [30] of the authors for the constant viscosities to the case of density dependent viscosities which are degenerate at vacuum states.

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

Section: Proof Of Theorem 21mentioning

confidence: 99%

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

Luo

Zeng

2015

Comm Pure Appl Math

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For the physical vacuum free boundary problem with the sound speed being C 1/2 -Hölder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the the porous media equation with the same total mass when the initial data is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in supreme norm and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates and elliptic estimates.

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Well‐posedness for compressible Euler equations with physical vacuum singularity

Cited by 146 publications

References 25 publications

A Priori Estimates for the Free-Boundary 3D Compressible Euler Equations in Physical Vacuum

A Priori Estimates for the Free-Boundary 3D Compressible Euler Equations in Physical Vacuum

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

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

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