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

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Abstract

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 \({\gamma}\) 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. Those uniform convergence are of fundamental importance in the study of vacuum free boundary problems which are missing in the previous results for global weak solutions. Moreover, the results obtained in this paper apply to much broader cases of viscosities than those in Fang and Zhang (Arch Ration Mech Anal 191:195–243, 2009) for the theory of weak solutions when the adiabatic exponent \({\gamma}\) lies in the most physically relevant range. Finally, this paper extends the previous local-in-time theory for strong solutions to a global-in-time one.

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Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Tao Luo

  2. Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

    Zhouping Xin

  3. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Huihui Zeng

  4. Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, 02318, USA

    Huihui Zeng

Authors
  1. Tao Luo
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  2. Zhouping Xin
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  3. Huihui Zeng
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Correspondence to Tao Luo.

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Communicated by H.-T. Yau

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Luo, T., Xin, Z. & Zeng, H. Nonlinear Asymptotic Stability of the Lane-Emden Solutions for the Viscous Gaseous Star Problem with Degenerate Density Dependent Viscosities. Commun. Math. Phys. 347, 657–702 (2016). https://doi.org/10.1007/s00220-016-2753-1

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