On the singular set and the uniqueness of weak solutions of the Navier- Stokes equations
type: Item
instance of: scholarly article
title: On the singular set and the uniqueness of weak solutions of the Navier- Stokes equations
zbMATH Open document ID: 0567.35069
DOI: 10.1007/BF01174870
author: Hermann Sohr
author: Wolf von Wahl
published in: Manuscripta Mathematica
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155034
review text: This paper establishes three results for weak solutions to the Navier- Stokes equations. Firstly it proves a uniqueness theorem by showing that if u, v are weak solutions with the same data, if \(u\in L^{\infty}((0,T),L^ n(\Omega))\) with \(\Omega\) bounded and if v satisfies the energy inequality everywhere then \(u=v\). Secondly, let u be any weak solution in the class \(L^{\infty}((0,T),L^ n(\Omega)).\) Then there are at most countably many points in [0,T] where \(\| u(t)\|_{L^ n(\Omega)}\) is discontinuous from the left while u is continuous from the right at each t in [0,T], in the \(L^ n(\Omega)\)- topology. Finally it is shown that for \(n=4\) any weak solution for which the energy inequality holds in regular with the exception of a bounded set of measure 0.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 3905113
zbMATH Keywords: singular set
zbMATH Keywords: weak solutions
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: uniqueness
zbMATH Keywords: energy inequality
reviewed by: Brian Straughan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3905113