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Blowup Criterion for Viscous Baratropic Flows with Vacuum States

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Abstract

We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional (3D) barotropic compressible Navier-Stokes equations. More precisely, if a solution of the 3D barotropic compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce’s criterion for 3-dimensional incompressible Euler equations (Ponce in Commun Math Phys 98:349–353, 1985). In addition, initial vacuum states are allowed in our cases.

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Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, 230026, P. R. China

    Xiangdi Huang

  2. Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing, 100190, P. R. China

    Jing Li

  3. Hua Loo-Keng Key Laboratory of Mathematics, Academia Sinica, Beijing, 100190, P. R. China

    Jing Li

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

    Xiangdi Huang, Jing Li & Zhouping Xin

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  1. Xiangdi Huang
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  2. Jing Li
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  3. Zhouping Xin
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Correspondence to Xiangdi Huang.

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Communicated by P. Constantin

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Huang, X., Li, J. & Xin, Z. Blowup Criterion for Viscous Baratropic Flows with Vacuum States. Commun. Math. Phys. 301, 23–35 (2011). https://doi.org/10.1007/s00220-010-1148-y

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