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

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Abstract

This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang and Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Georgetown University, Washington, D.C., USA

    Tao Luo

  2. Institute of Mathematical Sciences, Chinese University of Hong Kong, Hong Kong, China

    Zhouping Xin

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

    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 Huihui Zeng.

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Communicated by A. Bressan

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Luo, T., Xin, Z. & Zeng, H. Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional Compressible Euler Equations with or without Self-Gravitation. Arch Rational Mech Anal 213, 763–831 (2014). https://doi.org/10.1007/s00205-014-0742-0

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  • DOI: https://doi.org/10.1007/s00205-014-0742-0

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