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Stability of shock waves for the Broadwell equations

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Abstract

For the Broadwell model of the nonlinear Boltzmann equation, there are shock profile solutions, i.e. smooth traveling waves that connect two equilibrium states. For weak shock waves, we prove asymptotic (in time) stability with respect to small perturbations of the initial data. Following the work of Liu [7] on shock wave stability for viscous conservation laws, the method consists of analyzing the solution as the sum of a shock wave, a diffusive wave, a linear hyperbolic wave and an error term. The diffusive and linear hyperbolic waves are approximate solutions of the fluid dynamic equations corresponding to the Broadwell model. The error term is estimated using a variation of the energy estimates of Kawashima and Matsumura [6] and the characteristic energy method of Liu [7].

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, 251 Mercer Street, 10012, New York, NY, USA

    Russel E. Caflisch

  2. Mathematics Department, University of Maryland, 20742, College Park, MD, USA

    Tai-Ping Liu

Authors
  1. Russel E. Caflisch
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  2. Tai-Ping Liu
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Communicated by J. L. Lebowitz

Research supported by the Office of Naval Research through grant N00014-81-0002 and by the National Science Foundation through grant NSF-MCS-83-01260

Research supported by the National Science Foundation through grant DMS-84-01355

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Caflisch, R.E., Liu, TP. Stability of shock waves for the Broadwell equations. Commun.Math. Phys. 114, 103–130 (1988). https://doi.org/10.1007/BF01218291

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  • DOI: https://doi.org/10.1007/BF01218291

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