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Inventiones mathematicae

, Volume 206, Issue 3, pp 935–974 | Cite as

Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations

  • Alexis F. VasseurEmail author
  • Cheng Yu
Article

Abstract

In this paper, we prove the existence of global weak solutions for 3D compressible Navier–Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins (Commun Math Phys 238:211–223 2003) entropy conservation. The main contribution of this paper is to derive the Mellet and Vasseur (Commun Partial Differ Equ 32:431–452, 2007) type inequality for weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the compressible barotropic Navier–Stokes equations. The result holds for any \(\gamma >1\) in two dimensional space, and for \(1<\gamma <3\) in three dimensional space, in both case with large initial data possibly vanishing on the vacuum. This solves an open problem proposed by Lions (Mathematical topics in fluid mechanics. Vol. 2. Compressible models, 1998).

Mathematics Subject Classification

35Q35 76N10 

Notes

Acknowledgments

A. Vasseur’s research was supported in part by NSF Grant DMS-1209420. C. Yu’s research was supported in part by an AMS-Simons Travel Grant.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Texas at AustinAustinUSA

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