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Non-formation of vacuum states for Navier-Stokes equations with density-dependent viscosity

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Abstract

We consider the Cauchy problem, free boundary problem and piston problem for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Using the reduction to absurdity method, we prove that the weak solutions to these systems do not exhibit vacuum states, provided that no vacuum states are present initially. The essential requirements on the solutions are that the mass and energy of the fluid are locally integrable at each time, and the L loc 1-norm of the velocity gradient is locally integrable in time.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Zhang Ting & Fang Dao-yuan

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  1. Zhang Ting
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  2. Fang Dao-yuan
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Correspondence to Fang Dao-yuan.

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Project supported by the National Natural Science Foundation of China (No. 10571158) and the DFG

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Zhang, T., Fang, Dy. Non-formation of vacuum states for Navier-Stokes equations with density-dependent viscosity. J. Zhejiang Univ. - Sci. A 8, 1681–1690 (2007). https://doi.org/10.1631/jzus.2007.A1681

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  • DOI: https://doi.org/10.1631/jzus.2007.A1681

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