Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (1): 389-412.doi: 10.1016/S0252-9602(12)60025-3

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ASYMPTOTIC BEHAVIOR OF SOLUTIONS TOWARD THE SUPERPOSITION OF CONTACT DISCONTINUITY AND SHOCK WAVE FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

Hakho Hong, Feimin Huang   

  1. Institute of Mathematics, Academy of Sciences, Pyonyang, DPR of Korea|Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, China
  • Received:2011-12-15 Online:2012-01-20 Published:2012-01-20
  • Supported by:

    The first author’s research was partially supported by NSFC (10825102) for distinguished youth scholar. The second author’s research was supported by the CAS-TWAS postdoctoral fellowships (FR number: 3240223274) and AMSS in Chinese Academy of Sciences.

Abstract:

A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave is established under some smallness conditions. To do this, we first construct a new viscous contact wave such that the momentum equation is satisfied exactly and then determine the shift of the viscous shock wave. By using them
together with an inequality concerning the heat kernel in the half space, we obtain the desired a priori estimates. The proof is based on the elementary energy method by the anti-derivative argument.

Key words: compressible Navier-Stokes equations, free boundary, superposition of shock wave and contact discontinuity, stability

CLC Number: 

  • 35Q30
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