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

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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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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Keywords

  • Energy Estimate
  • Wave Pattern
  • Contact Discontinuity
  • Basic Wave
  • Excess Mass