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Uniqueness of a Planar Contact Discontinuity for 3D Compressible Euler System in a Class of Zero Dissipation Limits from Navier–Stokes–Fourier System

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Abstract

We prove the stability of a planar contact discontinuity without shear, a family of special discontinuous solutions for the three-dimensional full Euler system, in the class of vanishing dissipation limits of the corresponding Navier–Stokes–Fourier system. We also show that solutions of the Navier–Stokes–Fourier system converge to the planar contact discontinuity when the initial datum converges to the contact discontinuity itself. This implies the uniqueness of the planar contact discontinuity in the class that we are considering. Our results give an answer to the open question, whether the planar contact discontinuity is unique for the multi-D compressible Euler system. Our proof is based on the relative entropy method, together with the theory of a-contraction up to a shift and our new observations on the planar contact discontinuity.

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Acknowledgements

M.-J. Kang was partially supported by the NRF-2019R1C1C1009355. A. Vasseur was partially supported by the NSF Grant: DMS 1614918. Y. Wang is supported by NSFC Grants Nos. 12090014 and 11688101.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, 34141, Korea

    Moon-Jin Kang

  2. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712, USA

    Alexis F. Vasseur

  3. Institute of Applied Mathematics, AMSS, CAS, Beijing, 100190, People’s Republic of China

    Yi Wang

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

    Yi Wang

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  1. Moon-Jin Kang
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Correspondence to Moon-Jin Kang.

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Kang, MJ., Vasseur, A.F. & Wang, Y. Uniqueness of a Planar Contact Discontinuity for 3D Compressible Euler System in a Class of Zero Dissipation Limits from Navier–Stokes–Fourier System. Commun. Math. Phys. 384, 1751–1782 (2021). https://doi.org/10.1007/s00220-021-04100-3

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