Generic solvability of the equations of Navier-Stokes
type: Item
instance of: scholarly article
title: Generic solvability of the equations of Navier-Stokes
zbMATH Open document ID: 0676.35073
author: Hermann Sohr
author: Wolf von Wahl
published in: Hiroshima Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: The authors study the existence of unique strong solutions for arbitrary times \(T>0\) of the initial boundary value problem for the Navier-Stokes equations on bounded domains in \({\mathbb{R}}^ 3\). For given suitable initial data \(u_ 0\) they investigate the question whether any body force \(f\in L^ p(0,T;L^ p(\Omega)^ 3)\), \(p\in [2,\infty)\), \(\Omega\) being the domain in \({\mathbb{R}}^ 3\), leads to a unique strong solution for arbitrary \(T>0\). The set of such f is open in \(L^ p(0,T;L^ p(\Omega)^ 3)\). The authors prove in this paper that the set of these forces f lies dense in \(L^ p(0,T;L^ p(\Omega)^ 3)\) with respect to the norm of \(L^ s(0,T;L^ q(\Omega)^ 3)\) for all \(s,q\in (1,\infty)\) with \(4<2/s+3/q.\) Therefore, in this sense the set of forces f in \(L^ p(0,T;L^ p(\Omega)^ 3)\) forms a large subset, i.e. the problem is generically solvable.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 4107309
zbMATH Keywords: existence
zbMATH Keywords: unique
zbMATH Keywords: strong solutions
zbMATH Keywords: initial boundary value problem
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: initial data
zbMATH Keywords: generically solvable
reviewed by: Gerald Warnecke
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4107309