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Stability of Contact Discontinuities for the 1-D Compressible Navier-Stokes Equations

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Abstract

In this paper, we study the large-time asymptotic behavior of solutions of the one-dimensional compressible Navier-Stokes system toward a contact discontinuity, which is one of the basic wave patterns for the compressible Euler equations. It is proved that such a weak contact discontinuity is a metastable wave pattern, in the sense introduced in [24], for the 1-D compressible Navier-Stokes system for polytropic fluid by showing that a viscous contact wave, which approximates the contact discontinuity on any finite-time interval for small heat conduction and then runs away from it for large time, is nonlinearly stable with a uniform convergence rate provided that the initial excess mass is zero. This result is proved by an elaborate combination of elementary energy estimates with a weighted characteristic energy estimate, which makes full use of the underlying structure of the viscous contact wave.

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Authors and Affiliations

  1. Institute of Applied Mathematics AMSS, Academia Sinica, China

    Feimin Huang

  2. Department of Pure and Applied Mathematics Graduate School of Information Science and Technology, Osaka University, Japan

    Akitaka Matsumura

  3. The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong

    Zhouping Xin

Authors
  1. Feimin Huang
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  2. Akitaka Matsumura
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  3. Zhouping Xin
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Correspondence to Feimin Huang.

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Communicated by A. Bressan

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Huang, F., Matsumura, A. & Xin, Z. Stability of Contact Discontinuities for the 1-D Compressible Navier-Stokes Equations. Arch. Rational Mech. Anal. 179, 55–77 (2006). https://doi.org/10.1007/s00205-005-0380-7

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  • DOI: https://doi.org/10.1007/s00205-005-0380-7

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