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Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows

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Abstract

This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin’s condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.

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Authors and Affiliations

  1. NCMIS, AMSS, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Xiangdi Huang

  2. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Osaka, 565-0871, Japan

    Xiangdi Huang

  3. Institute of Applied Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Jing Li

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  1. Xiangdi Huang
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  2. Jing Li
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Correspondence to Jing Li.

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Communicated by L. Caffarelli

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Huang, X., Li, J. Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows. Commun. Math. Phys. 324, 147–171 (2013). https://doi.org/10.1007/s00220-013-1791-1

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  • DOI: https://doi.org/10.1007/s00220-013-1791-1

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