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Vanishing Viscosity Limit to the Planar Rarefaction Wave for the Two-Dimensional Compressible Navier–Stokes Equations

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Abstract

The vanishing viscosity limit of the two-dimensional (2D) compressible and isentropic Navier–Stokes equations is studied in the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. It is proved that there exists a family of smooth solutions for the 2D compressible Navier–Stokes equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficients away from the initial time. In the proof, the hyperbolic wave is crucially introduced to recover the physical viscosities of the inviscid rarefaction wave profile, in order to rigorously justify the vanishing viscosity limit.

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Acknowledgements

The research of D. Wang is partially supported by the National Science Foundation under Grants DMS-1312800 and DMS-1613213. Y. Wang is supported by NSFC Grant Nos. 11671385 and 11688101 and CAS Interdisciplinary Innovation Team. The authors would like to thank the anonymous referee for the valuable suggestions which improved the presentation.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Lin-An Li & Yi Wang

  2. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Dehua Wang

  3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Lin-An Li & Yi Wang

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  1. Lin-An Li
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  2. Dehua Wang
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  3. Yi Wang
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Correspondence to Dehua Wang.

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Communicated by C. De Lellis

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Li, LA., Wang, D. & Wang, Y. Vanishing Viscosity Limit to the Planar Rarefaction Wave for the Two-Dimensional Compressible Navier–Stokes Equations. Commun. Math. Phys. 376, 353–384 (2020). https://doi.org/10.1007/s00220-019-03580-8

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  • DOI: https://doi.org/10.1007/s00220-019-03580-8

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