Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in \(\mathbb{R}^ 3\)
type: Item
instance of: scholarly article
title: Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in \(\mathbb{R}^ 3\)
zbMATH Open document ID: 0865.35101
DOI: 10.1016/S0294-1449(16)30107-X
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1996-12-12T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78385
review text: The author investigates the global solvability of the incompressible Navier-Stokes equation in \(\mathbb{R}^3\). For that purpose he introduces the homogeneous Besov spaces \(\dot B^\alpha_{p,q}\). The author proves that a global solution with initial data \(u_0\) exists and is unique in the space \(H^s(\mathbb{R}^3)\), \(s>1/2\), if \(|u_0|_{\dot B^{-1/4}_{4,\infty}}\) is smaller than some universal constant, and the solution is in \(L^p\), \(p>3/2\), if \(u_0\in L^p\cap\dot B^{-(1-3/2p)}_{2p,\infty}\) and \(|u_0|_{\dot B^{-(1-3/2p)}_{2p,\infty}}\) is smaller than some constant depending on \(p\). In the latter result \(u_0\in L^p\cap\dot B^{-(1-3/2p)}_{2p,\infty}\) may be replaced by \(u_0\in L^p\cap L^3\), and if \(p>3\) by \(L^2\cap L^p\). Some regularity results of solutions are also established. In the proofs, the author transforms the original problem into an integral equation, and uses the exact expression of the heat kernel instead of the semigroup theory to obtain the necessary estimates.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 58D25
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 939890
zbMATH Keywords: Besov space
zbMATH Keywords: heat kernel
zbMATH Keywords: paraproduct decomposition
zbMATH Keywords: Bernstein's lemma
author: Fabrice Planchon
reviewed by: Hiroki Tanabe
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 939890