Nonlinear Stability of Rarefaction Waves for the Boltzmann Equation |
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Authors: | Tai-Ping Liu Tong Yang Shih-Hsien Yu Hui-Jiang Zhao |
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Institution: | (1) Institute of Mathematics, Academia Sinica, Taipei;(2) Department of Mathematics, Stanford University,;(3) Department of Mathematics, City University of Hong Kong,;(4) School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China;(5) School of Political Science and Economics, Waseda University, Tokyo 169-8050, Japan |
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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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