Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional Compressible Euler Equations with or without Self-Gravitation

Luo, Tao; Xin, Zhouping; Zeng, Huihui

doi:10.1007/s00205-014-0742-0

Cited by 103 publications

(78 citation statements)

References 44 publications

(96 reference statements)

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“…[5,11]) and continuation arguments. The uniqueness of the smooth solutions can be obtained as in section 11 of [20]. In order to prove the global existence of smooth solutions, we need to obtain the uniformin-time a priori estimates on any given time interval…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…A nice review of singular behavior of solutions near vacuum boundaries for compressible fluids can be found in [21]. In [19], a general uniqueness theorem was proved for three-dimensional motions of compressible Euler equations with or without self-gravitation, and a new local-in-time well-posedness theory was established for spherically symmetric motions without imposing the compatibility condition of the first derivative being 0 at the center of symmetry. An instability theory of stationary solutions to the physical vacuum free boundary problem for the spherically symmetric compressible Euler-Poisson equations of gaseous stars for 6=5 < < 4=3 was established in [10].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Proof Of Theorem 21mentioning

confidence: 99%

Section: Introductionmentioning

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 spherically symmetric EP γ -system local well-posedness is implicit in the work of Jang [24] and it was also shown by Luo, Xin, & Zeng [35]. Finally, when the underlying domain inherits the topology of the manifold T 2 × R with a coupling to the force field given via the convolution kernel 1 |·| , Gu & Lei [11] showed local well-posedness relying on the framework developed in [4].…”

mentioning

confidence: 78%

A Class of Global Solutions to the Euler–Poisson System

Hadžić

Jang

2019

Commun. Math. Phys.

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Using recent developments in the theory of globally defined expanding compressible gases, we construct a class of global-in-time solutions to the compressible 3-D Euler-Poisson system without any symmetry assumptions in both the gravitational and the plasma case. Our allowed range of adiabatic indices includes, but is not limited to all γ of the form γ = 1 + 1 n , n ∈ N \ {1}. The constructed solutions have initially small densities and a compact support. As t → ∞ the density scatters to zero and the support grows at a linear rate in t.The three dimensional compressible Euler-Poisson system couples the equation for a compressible gas to a self-consistent force field created by the gas particles: if the interaction is gravitational, we refer to the model as the gravitational Euler-Poisson system and if the interaction is electrostatic we talk about the electrostatic Euler-Poisson system. In the gravitational case we obtain a model of a Newtonian star [45,1,2], while in the case of repelling forces between the particles, we arrive at a model for plasmas [12,14]. We shall work with the free-boundary formulation of the problem, wherein a moving *

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“…These tools have been first designed and successfully used to overcome the above mentioned difficulties in the context of compressible Euler equations in vacuum [8,22] and in the context of the Euler-Poisson system in [17,18]. A similar methodology was used to prove local-in-time existence for the Euler-Poisson system [26]. In [30], Nash-Moser theory was used to establish the existence of smooth local solutions of the Euler-Poisson system approximating time periodic (linearized) profiles.…”

Section: Comments and Methodologymentioning

confidence: 99%

Nonlinear Stability of Expanding Star Solutions of the Radially Symmetric Mass‐Critical Euler‐Poisson System

Hadžić

Jang

2017

Comm Pure Appl Math

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We prove nonlinear stability of compactly supported expanding star solutions of the mass‐critical gravitational Euler‐Poisson system. These special solutions were discovered by Goldreich and Weber in 1980. The expansion rate of such solutions can be either self‐similar or non‐self‐similar (linear), and we treat both types. An important outcome of our stability results is the existence of a new class of global‐in‐time radially symmetric solutions, which are not homologous and therefore not encompassed by the existing works. Using Lagrangian coordinates we reformulate the associated free‐boundary problem as a degenerate quasilinear wave equation on a compact spatial domain. The problem is mass‐critical with respect to an invariant rescaling and the analysis is carried out in similarity variables. © 2017 Wiley Periodicals, Inc.

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Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional Compressible Euler Equations with or without Self-Gravitation

Cited by 103 publications

References 44 publications

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

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

A Class of Global Solutions to the Euler–Poisson System

Nonlinear Stability of Expanding Star Solutions of the Radially Symmetric Mass‐Critical Euler‐Poisson System

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