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Nonlinear Stability of Rarefaction Waves for the Boltzmann Equation

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Abstract

It is well known that the Boltzmann equation is related to the Euler and Navier-Stokes equations in the field of gas dynamics. The relation is either for small Knudsen number, or, for dissipative waves in the time-asymptotic sense. In this paper, we show that rarefaction waves for the Boltzmann equation are time-asymptotic stable and tend to the rarefaction waves for the Euler and Navier-Stokes equations. Our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro-macro decomposition of the Boltzmann equation. The expansion nature of the rarefaction waves and the suitable microscopic version of the H-theorem are essential elements of our analysis.

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Authors and Affiliations

  1. Institute of Mathematics, Academia Sinica, Taipei, Taiwan

    Tai-Ping Liu

  2. Department of Mathematics, Stanford University, USA

    Tai-Ping Liu

  3. Department of Mathematics, City University of Hong Kong, USA

    Tong Yang & Shih-Hsien Yu

  4. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Hui-Jiang Zhao

  5. School of Political Science and Economics, Waseda University, Tokyo, 169-8050, Japan

    Hui-Jiang Zhao

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  1. Tai-Ping Liu
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  2. Tong Yang
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  4. Hui-Jiang Zhao
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Correspondence to Tai-Ping Liu.

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Liu, TP., Yang, T., Yu, SH. et al. Nonlinear Stability of Rarefaction Waves for the Boltzmann Equation. Arch Rational Mech Anal 181, 333–371 (2006). https://doi.org/10.1007/s00205-005-0414-1

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