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Finite Time Blow-Up of Regular Solutions for Compressible Flows

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Abstract

The development of finite time singularity of smooth solutions to the compressible Navier-Stokes system as well as its blowup mechanism is discussed in the presence of vacuum. It is shown that any smooth solutions to the compressible Navier-Stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support or isolated mass group. Besides, unified Serrin-type regularity criteria are established for the barotropic and full compressible Navier-Stokes equations with or without heat conduction. As an immediate corollary, it gives an affirmative answer to a problem proposed by J. Nash in the 1950s which asserts that the finite time blowup must be due to the concentration of either the density or the temperature.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, AMSS, Chinese Academy of Sciences, 100190, Beijing, China

    Xiangdi Huang

  2. The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Room 701 Academic Building No.1, Shatin, Hong Kong, China

    Xiangdi Huang & Zhou Ping Xin

Authors
  1. Xiangdi Huang
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  2. Zhou Ping Xin
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Correspondence to Xiangdi Huang .

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Editors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan

    Yoshikazu Giga

  2. Université de Toulon IMATH, Toulon, France

    Antonín Novotný

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Huang, X., Xin, Z. (2018). Finite Time Blow-Up of Regular Solutions for Compressible Flows. In: Giga, Y., Novotný, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-13344-7_57

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