Bilinear estimates in \(BMO\) and the Navier-Stokes equations
type: Item
instance of: scholarly article
title: Bilinear estimates in \(BMO\) and the Navier-Stokes equations
zbMATH Open document ID: 0970.35099
author: Hideo Kozono
author: Yasushi Taniuchi
published in: Mathematische Zeitschrift
publication date: 2001-10-10T00:00:00Z
review text: The authors investigate blow-up phenomena of smooth solutions to the Cauchy problem for the Navier-Stokes system in \({\mathbb R}^n\), \((n\geq 3)\): \[ u_t-\Delta u+u\cdot \nabla u +\nabla p =0,\quad \text{ div} u =0,\quad u(x,0)=a(x). \] It is well known that for every initial datum \(a\in W^{s,2}({\mathbb R}^n)\), \(s>n/2-1\), there exists \(T=T(\|a\|_{W^{s,2}})\) and a unique regular solution \(u(x,t)\) to this Cauchy problem on the interval \([0,T)\). Then there is the important question whether the solution loses its regularity at \(t=T\). \textit{Y. Giga} [J. Differ. Equations 61, 186-212 (1986; Zbl 0577.35058)] showed that if \(\int_0^T \|u(t)\|^\kappa_{L^r} dt<\infty\) for \(2/\kappa +n/r=1\) with \(n<r< \infty\), then \(u\) can be continued beyond \(t=T\). Moreover, in \({\mathbb R}^3\), \textit{J. T. Beale, T. Kato} and \textit{A. Majda} [Commun. Math. Phys. 94, 61-66 (1984; Zbl 0573.76029)] dealt with the vorticity \(\omega=\text{ rot} u\) and proved an analogous result under the condition \(\int_0^T \|\omega (t)\|_{L^\infty} dt<\infty.\) The purpose of this paper is to extend these results to the marginal space \(BMO\) (which is larger than \(L^\infty\)), and it was proved that solutions can be extended beyond \(t=T\) provided \(\int_0^T \|u (t)\|^2_{BMO} dt <\infty.\) Moreover, the authors characterize the blow-up time by means of the vorticity \(\omega\) in \(BMO\) obtaining the result analogous to the one by Beal, Kato and Majda with \(L^\infty\) replaced by \(BMO\).
reviewed by: Grzegorz Karch
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 1533439
zbMATH Keywords: Navier-Stokes system
zbMATH Keywords: blow-up phenomena
zbMATH Keywords: smooth solutions
zbMATH Keywords: marginal space BMO
zbMATH Keywords: Cauchy problem
MaRDI profile type: Publication
full work available at URL: s002090000130
OpenAlex ID: W2017780476
DOI: 10.1007/S002090000130
Recommended article: Q3371947
Recommended article: On the critical one component regularity for 3-D Navier-Stokes system
Recommended article: Remark on the blow-up criterion of strong solutions to the Navier-Stokes equations in multiplier spaces
Recommended article: Critical one component anisotropic regularity for 3-DNavier-Stokes system
Recommended article: Blow-up solutions appearing in the vorticity dynamics with linear strain
Recommended article: On the finite-time singularities of the 3D incompressible Euler equations
Recommended article: A necessary condition of potential blowup for the Navier-Stokes system in half-space
Recommended article: Navier–Stokes Blowup Rates in Certain Besov Spaces Whose Regularity Exceeds the Critical Value by \({\boldsymbol{\epsilon \in [1,2]}}\)
Recommended article: A Blow-Up Criterion for the Nonhomogeneous Incompressible Navier--Stokes Equations
Recommended article: Q3119559
description: scientific article; zbMATH DE number 1533439