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Weak Solutions for the Compressible Navier-Stokes Equations with Density Dependent Viscosities

In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Author

Listed:
  • Didier Bresch

    (Université de Savoie Mont-Blanc, LAMA UMR 5127 CNRS Batiment le Chablais)

  • Benoît Desjardins

    (CMLA, ENS Cachan, CNRS and Modélisation Mesures et Applications S.A., Fondation Mathématique Jacques Hadamard)

Abstract

In this chapter, we focus on compressible Navier-Stokes equations with density-dependent viscosities in the multidimensional space case. The main objective of these notes is to present at the level of beginners an introduction to such systems showing the difference with the constant viscosity case. The guideline is to show a nonlinear hypercoercivity property due to the density dependency of the viscosities, to explain how it may be used to provide global existence of weak solutions to the barotropic compressible Navier-Stokes equations, and to the heat-conducting Navier-Stokes equations with a total energy formulation. We will also focus on the relative entropy method for such systems showing the difficulty coming from the density dependency. We hope to motivate by this chapter young researchers to work on such difficult topic trying to fill the gap between the constant viscosities case and the density-dependent viscosities satisfying the BD relation, trying to relax some modeling hypotheses and to extend the results.

Suggested Citation

  • Didier Bresch & Benoît Desjardins, 2018. "Weak Solutions for the Compressible Navier-Stokes Equations with Density Dependent Viscosities," Springer Books, in: Yoshikazu Giga & Antonín Novotný (ed.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, chapter 30, pages 1547-1599, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-13344-7_44
    DOI: 10.1007/978-3-319-13344-7_44
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