On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations
type: Item
instance of: scholarly article
title: On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations
zbMATH Open document ID: 0965.35121
DOI: 10.4171/RMI/268
author: Marco Cannone
author: Fabrice Planchon
published in: Revista Matemática Iberoamericana
publication date: 2001-07-19T00:00:00Z
full work available at URL: https://eudml.org/doc/39588
review text: The authors consider the Cauchy problem for the Navier-Stokes equations by reducing to the integral equation \[ u(x,t)=S(t) u_{0}(x)-\int_{0}^{t}\mathbf{P} S(t-s) \nabla \cdot (u \otimes u) (x,s) ds,\tag{1} \] where \(S(t)=\exp (t \Delta)\) is the heat kernel, \(\mathbf{P}\) is a pseudo-differential operator. Let \(u(x,t)\) be a solution of (1) in \(L^3 (\mathbb{R})\) on \(x\) and in \(C[0,T)\) on \(t\) with initial data \(u_0 \in L^3\), \(w:=u-S(t) u_0\), then \(w \in H \times C[0,T)\). Here \(H\) is a Sobolev space.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D03
zbMATH DE Number: 1485603
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: Besov space
reviewed by: Vladimir Mityushev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1485603