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Relative Entropies, Suitable Weak Solutions, and Weak-Strong Uniqueness for the Compressible Navier–Stokes System

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Abstract

We introduce the notion of relative entropy for the weak solutions to the compressible Navier–Stokes system. In particular, we show that any finite energy weak solution satisfies a relative entropy inequality with respect to any couple of smooth functions satisfying relevant boundary conditions. As a corollary, we establish the weak-strong uniqueness property in the class of finite energy weak solutions, extending thus the classical result of Prodi and Serrin to the class of compressible fluid flows.

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Authors and Affiliations

  1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Zitná 25, 115 67, Praha 1, Czech Republic

    Eduard Feireisl

  2. Department of Mathematics Education, Mokpo National University, Muan, 534-729, South Korea

    Bum Ja Jin

  3. IMATH Université du Sud Toulon-Var, BP 132, 83957, La Garde, France

    Antonín Novotný

Authors
  1. Eduard Feireisl
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  2. Bum Ja Jin
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  3. Antonín Novotný
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Correspondence to Eduard Feireisl.

Additional information

Communicated by D. Hoff

Eduard Feireisl was supported and the third author was partially supported by Grant 201/09/0917 of GA ČR as a part of the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503. Bum Ja Jin was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2011-0007701).

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Feireisl, E., Jin, B.J. & Novotný, A. Relative Entropies, Suitable Weak Solutions, and Weak-Strong Uniqueness for the Compressible Navier–Stokes System. J. Math. Fluid Mech. 14, 717–730 (2012). https://doi.org/10.1007/s00021-011-0091-9

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  • DOI: https://doi.org/10.1007/s00021-011-0091-9

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