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

By Alexis Vasseur, Cheng Yu. 2015

225 citations recorded by the source corpus.

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 $$\gamma >1$$γ>1 in two dimensional space, and for $$1<\gamma <3$$1<γ<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).

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/da82cd017a0b6154f45175973cfbcbac30c2577b

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities (2015, background): “uture. Remark 1.5 Around the same time when this paper is ﬁnished, there are announcements of some existence results on the problem (1.1) (1.3) (1.6) (1.7) with α= 1 and Ω = TN(N = 2,3) by Vasseur-Yu [38] with a diﬀerent approach. However, we have diﬃculties to understand some of their key a priori ”
Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor (2015, background): “The almost only successful insight in this case can be found in [15, 16, 17, 64, 48] with no dependency with respect 16”