Brezis-Gallouet-Wainger type inequalities and blow-up criteria for Navier-Stokes equations in unbounded domains
type: Item
instance of: scholarly article
title: Brezis-Gallouet-Wainger type inequalities and blow-up criteria for Navier-Stokes equations in unbounded domains
zbMATH Open document ID: 1397.35176
author: Kohei Nakao
author: Yasushi Taniuchi
published in: Communications in Mathematical Physics
publication date: 2018-05-25T00:00:00Z
review text: The authors consider the Navier-Stokes equations in a smooth domain \(\Omega \subset \mathbb R^3\). The initial condition is posed on the velocity \(u\) and \(u=0\) on \(\partial \Omega\). If the smooth solution \(u \in C([0, T)\) breaks down at a finite time \(t = T\), then \(\int_0^t ||rot \; u (\tau)||_{L^{\infty}} d\tau \nearrow \infty\) as \(t \nearrow T\). The authors discuss similar break down criteria in various spaces. The largest normed space where the \(\mathrm{rot }u\) can break down is found under some additional assumptions.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35A23
zbMATH DE Number: 6873210
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: break down criterion
zbMATH Keywords: Brezis-Gallouet-Wainger type inequality
reviewed by: Vladimir Mityushev
MaRDI profile type: Publication
full work available at URL: s00220-017-3061-0
OpenAlex ID: W2776427052
cites work: Remarks on the breakdown of smooth solutions for the 3-D Euler equations
cites work: \(L^ 2\) decay for the Navier-Stokes flow in halfspaces
cites work: On the semigroup of the Stokes operator for exterior domains in L q- spaces
cites work: An algebraic reduction of the 'scaling gap' in the Navier-Stokes regularity problem
cites work: Nonlinear Schrödinger evolution equations
cites work: A note on limiting cases of sobolev embeddings and convolution inequalities
cites work: Partial regularity of suitable weak solutions of the navier-stokes equations
cites work: On the well‐posedness of the Euler equations in the Triebel‐Lizorkin spaces
cites work: Q4210480
cites work: Q4207245
cites work: Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations
cites work: On the Stokes and Navier-Stokes System for Domains with Noncompact Boundary in Lq-spaces
cites work: HELMHOLTZ DECOMPOSITION AND STOKES RESOLVENT SYSTEM FOR APERTURE DOMAINS IN Lq-SPACES
cites work: On the blow-up of solutions of the 3-D Euler equations in a bounded domain
cites work: Q4167065
cites work: Analyticity of the semigroup generated by the Stokes operator in \(L_r\) spaces
cites work: Solutions in \(L_ r\) of the Navier-Stokes initial value problem
cites work: A regularity criterion for the 3D NSE in a local version of the space of functions of bounded mean oscillations
cites work: Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations
cites work: Double logarithmic inequality with a sharp constant
cites work: \(L_ q-L_ r\) estimates for solutions of the nonstationary Stokes equation in an exterior domain and the Navier-Stokes initial value problems in \(L_ q\) spaces
cites work: Strong \(L^ p\)-solutions of the Navier-Stokes equation in \(R^ m\), with applications to weak solutions
cites work: Commutator estimates and the euler and navier-stokes equations
cites work: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations
cites work: Limiting case of the Sobolev inequality in BMO, with application to the Euler equations
cites work: Remarks on Gagliardo-Nirenberg type inequality with critical Sobolev space and BMO
cites work: Q5424214
cites work: On the dissipative scale for the Navier-Stokes equation
cites work: Interpolation inequality of logarithmic type in abstract Besov spaces and an application to semilinear evolution equations
cites work: On nonstationary solutions of the Navier-Stokes equations in an exterior domain
cites work: Brézis-Gallouët-Wainger type inequality with a double logarithmic term in the Hölder space: its sharp constants and extremal functions
cites work: Sharp Sobolev Inequality of Logarithmic Type and the Limiting Regularity Condition to the Harmonic Heat Flow
cites work: A note on blow-up criterion to the 3-D Euler equations in a bounded domain
cites work: On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain.
cites work: On critical cases of Sobolev's inequalities
cites work: An extension of the Beale-Kato-Majda criterion for the Euler equations
cites work: A continuation principle for the 3-D Euler equations for incompressible fluids in a bounded domain
cites work: Q4273991
cites work: Q3142876
cites work: Q3173253
cites work: A regularity class for the Navier-Stokes equations in Lorentz spaces
cites work: Pseudodifferential operators and nonlinear PDE
cites work: Incompressible flows of an ideal fluid with vorticity in borderline spaces of besov type1
cites work: The Navier-Stokes initial value problem in \(L^ p\).
cites work: Uniqueness theorem for the basic nonstationary problem in the dynamics on an ideal incompressible fluid
cites work: Q4014109
DOI: 10.1007/S00220-017-3061-0
description: scientific article; zbMATH DE number 6873210