Existence of regular solutions to the stationary Navier-Stokes equations
type: Item
instance of: scholarly article
title: Existence of regular solutions to the stationary Navier-Stokes equations
zbMATH Open document ID: 0861.35074
DOI: 10.1007/BF01444513
author: Jens Frehse
author: M. Růžička
published in: Mathematische Annalen
publication date: 1995-11-08T00:00:00Z
full work available at URL: https://eudml.org/doc/165349
review text: With this paper the authors continue their interesting investigations on the regularity of solutions to the stationary Navier-Stokes equations [cf. Ann. Sc. Norm. Sup. Pisa 21, 63-95 (1994; Zbl 0823.35141) and Acta Appl. Math. 37, 53-66 (1994; Zbl 0814.35094)]. Starting from potential estimates for \({1\over 2}u^2+p\) and \((u(x)\cdot(x-x_0))^2\) from their previous work, where \((u,p)\) is a weak solution with periodic boundary conditions in \(N\) dimensions, \(5\leq N<10\), they first show that the potential of \((u(x)\cdot(x-x_0))^2\), if restricted to a suitably small ball \(B(x_0;r)\), becomes uniformly small with respect to \(x_0\) for all solutions \(u\) that are in a bounded set in \(W^{1,2}\). While the methods to show this inequality exploit the structure of the nonlinearity \((u\cdot\nabla)u\) in a delicate way the estimates of the Hölder seminorms of \(u\) and \(p\) are proven with arguments from the regularity theory for nonlinear elliptic systems. Finally, existence of regular solutions that belong to \(W^{2,q}\times W^{1,q} \forall q<\infty\) is shown by a degree argument, which can be applied because of the bounds on the Hölder norms. [For \(N=5\) the existence of regular solutions was also obtained by \textit{M. Struwe} in ibid. 719-741 (1995; reviewed below)].
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 800585
zbMATH Keywords: regularity of solutions to the stationary Navier-Stokes equations
zbMATH Keywords: potential estimates
zbMATH Keywords: structure of the nonlinearity
zbMATH Keywords: degree argument
reviewed by: Josef Bemelmans
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 800585