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Finite Time Blow-Up of Regular Solutions for Compressible Flows

In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Author

Listed:
  • Xiangdi Huang

    (Chinese Academy of Sciences, Institute of Mathematics, AMSS
    The Chinese University of Hong Kong, The Institute of Mathematical Sciences)

  • Zhou Ping Xin

    (The Chinese University of Hong Kong, The Institute of Mathematical Sciences)

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.

Suggested Citation

  • Xiangdi Huang & Zhou Ping Xin, 2018. "Finite Time Blow-Up of Regular Solutions for Compressible Flows," Springer Books, in: Yoshikazu Giga & Antonín Novotný (ed.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, chapter 40, pages 2183-2261, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-13344-7_57
    DOI: 10.1007/978-3-319-13344-7_57
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