Global solutions to physical vacuum problem of non-isentropic viscous gaseous stars and nonlinear asymptotic stability of stationary solutions

Hong, Guangyi; Luo, Tao; Zhu, Changjiang

doi:10.1016/j.jde.2018.02.027

Cited by 15 publications

(5 citation statements)

References 42 publications

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“…[3] offered different a priori estimates in conormal Sobolev spaces for the local-in-time solutions, with fewer compatibility conditions but under the the physical vacuum condition in the isentropic case (i.e., (1.10) with δ " 1{pγ ´1q). Also see some works on other models of non-isentropic flows in [19,11,43,48,46,47].…”

Section: 2mentioning

confidence: 99%

Immediate blowup of classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier--Stokes equations

Liu¹,

Yuan²

2022

Preprint

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This paper considers the immediate blowup of classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier-Stokes equations, where the viscosities and the heat conductivity could be constants, or more physically, the degenerate, temperature-dependent functions which vanish on the vacuum boundary (i.e., µ " μθ α , λ " λθ α , κ " κθ α , for constants 0 ď α ď 1{pγ ´1q, μ ą 0, 2μ `nλ ě 0, κ ě 0, and adiabatic exponent γ ą 1). For such vacuum free boundary problems, imposing proper decaying or singular conditions on the density or temperature profiles across the vacuum boundary is important to the existence of the classical solutions.In our previous study (Liu and Yuan, Math. Models Methods Appl. Sci. ( 9) 12, 2019 ), with three-dimensional spherical symmetry and constant shear viscosity, vanishing bulk viscosity and heat conductivity, we established a class of globalin-time large solutions, with bounded entropy and entropy derivatives, under the condition the decaying rate of the initial density to the vacuum boundary is of any positive power of the distance function to the boundary. In this paper we prove that such classical solutions do not exist for any small time for non-vanishing bulk viscosity, provided the initial velocity is expanding near the boundary.When the heat conductivity does not vanish, it is automatically satisfied that the normal derivative of the temperature of the classical solution across the free boundary does not degenerate, which supports that such singular condition imposed on the initial data in our previous local well-posedness study (Liu and Yuan, SIAM J. Math. Anal. (2) 51, 2019 ) is physical; meanwhile, the entropy of the classical solution immediately blowups if the decaying rate of the initial density is not of 1{pγ ´1q power of the distance function to the boundary.We also have a similar non-existence result for the one-dimensional case, but with more restrictions on the parameter decaying rate of the initial density. These results provide a first-step study of the well-posedness theory of the vacuum free boundary problem of the viscous gas with degenerate, temperature-dependent transport coefficients. The proofs are obtained by the analyzing the boundary behaviors of the velocity, entropy and temperature, and investigating the maximum principles for parabolic equations with degenerate coefficients.

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Section: 2mentioning

confidence: 99%

Immediate blowup of classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier--Stokes equations

Liu¹,

Yuan²

2022

Preprint

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“…From ( 14) and ( 2), D = 0 if and only if u ∝ r in (0, R(t)). Therefore, unless we adopt a perturbation φ with φ ∝ r, the physical energy E defined in (14) decreases over time. For the instability theory, we will show that, in contrast to [13], the vanishing of the initial energy is not enough to guarantee the stability.…”

Section: Instability Of Self-similarly Expanding Homogeneous Solution...mentioning

confidence: 99%

On the expanding configurations of viscous radiation gaseous stars: the isentropic model

Liu

2019

Nonlinearity

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In this work, we study the stability of the expanding configurations for radiation gaseous stars. Such expanding configurations exist for the isentropic model and the thermodynamic model. In particular, we focus on the effect of the monatomic gas viscosity tensor on the expanding configurations. With respect to small perturbation, it is shown the self-similarly expanding homogeneous solutions of the isentropic model is unstable, while the linearly expanding homogeneous solutions of the isentropic model and the thermodynamic model are stable. This is an extensive study of the result in arXiv:1605.08083 by Hadzic and Jang.MSC class: 35A01, 35Q30, 35Q35, 35Q85, 76E20

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“…They applied weighted energy methods to overcome the extreme difficulties caused by boundary singularity or coordinate singularity. For more global existence results of viscous flow with free boundary, we refer to [9,21,34,35,38,39,43] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On Vacuum Free Boundary Problem of the Spherically Symmetric Euler Equations with Damping and Solid Core

Wang¹

2022

Preprint

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In this paper, the global existence of smooth solution and the long-time asymptotic stability of the equilibrium to vacuum free boundary problem of the spherically symmetric Euler equations with damping and solid core have been obtained for arbitrary finite positive gas constant A in the state equation p = Aρ γ with p being the pressure and ρ the density, provided that γ > 4/3, initial perturbation is small and the radius of the equilibrium R is suitably larger than the radius of the solid core r 0 . Moreover, we obtain the pointwise convergence from the smooth solution to the equilibrium in a surprisingly exponential time-decay rate. The proof is mainly based on weighted energy method in Lagrangian coordinate.

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Global solutions to physical vacuum problem of non-isentropic viscous gaseous stars and nonlinear asymptotic stability of stationary solutions

Cited by 15 publications

References 42 publications

Immediate blowup of classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier--Stokes equations

Immediate blowup of classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier--Stokes equations

On the expanding configurations of viscous radiation gaseous stars: the isentropic model

On Vacuum Free Boundary Problem of the Spherically Symmetric Euler Equations with Damping and Solid Core

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