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

• 成果类型: Article
• 署名作者: Vasseur, Alexis F.; Yu, Cheng
• 署名单位: University of Texas System; University of Texas Austin
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0666-4
• 发表日期: 2016
• 页码: 935-974
• 关键词: discontinuous initial data; shallow-water equations; boundary-value-problems; critical spaces; multidimensional flows; viscous fluids; well-posedness; CONVERGENCE; uniqueness; density

摘要

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 in two dimensional space, and for 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).