On the Navier-Stokes equations in scaling-invariant spaces in any dimension
type: Item
instance of: scholarly article
title: On the Navier-Stokes equations in scaling-invariant spaces in any dimension
zbMATH Open document ID: 1410.35144
author: Kazuo Yamazaki
published in: Revista Matemática Iberoamericana
publication date: 2019-02-14T00:00:00Z
review text: Summary: We study the Navier-Stokes equations with a dissipative term that is generalized through a fractional Laplacian in any dimension higher than two. We extend the horizontal Biot-Savart law beyond dimension three. Using the anisotropic Littlewood-Paley theory with which we distinguish the first two directions from the rest, we obtain a blow-up criteria for its solution in norms which are invariant under the rescaling of these equations. The proof goes through for the classical Navier-Stokes equations if dimension is three, four or five. We also give heuristics and partial results toward further improvement.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 7023569
zbMATH Keywords: anisotropic Littlewood-Paley theory
zbMATH Keywords: blow-up
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: regularity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/rmi/1034
OpenAlex ID: W2908105684
cites work: Fourier Analysis and Nonlinear Partial Differential Equations
cites work: Remarks on the breakdown of smooth solutions for the 3-D Euler equations
cites work: A new regularity class for the Navier-Stokes equations in \(\mathbb{R}^ n\)
cites work: The spaces \(L^ p\), with mixed norm
cites work: Regularity criteria for the three-dimensional Navier-Stokes equations
cites work: Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor
cites work: Incompressible flow in porous media with fractional diffusion
cites work: Q4242041
cites work: Q4210480
cites work: On the critical one component regularity for 3-D Navier-Stokes system
cites work: On the critical one component regularity for 3-D Navier-Stokes system: general case
cites work: Partial regularity of solutions to the four-dimensional Navier-Stokes equations at the first blow-up time
cites work: The Navier-Stokes equations in the critical Lebesgue space
cites work: On partial regularity of steady-state solutions to the 6D Navier-Stokes equations
cites work: L3,∞-solutions of the Navier-Stokes equations and backward uniqueness
cites work: On the Navier-Stokes initial value problem. I
cites work: Q3843833
cites work: The resolution of the Navier-Stokes equations in anisotropic spaces
cites work: The Two-Dimensional Incompressible Boussinesq Equations with General Critical Dissipation
cites work: Strong \(L^ p\)-solutions of the Navier-Stokes equation in \(R^ m\), with applications to weak solutions
cites work: Blow up and regularity for fractal Burgers equation
cites work: One component regularity for the Navier–Stokes equations
cites work: Navier-Stokes equations with regularity in one direction
cites work: On the critical one-component velocity regularity criteria to 3-D incompressible MHD system
cites work: On anisotropic regularity criteria for the solutions to 3D Navier-Stokes equations
cites work: Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension
cites work: The Navier-Stokes equations in space dimension four
cites work: On the interior regularity of weak solutions of the Navier-Stokes equations
cites work: Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation
cites work: Generalized MHD equations.
cites work: The generalized incompressible Navier-Stokes equations in Besov spaces
cites work: Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces
cites work: Global regularity for a class of generalized magnetohydrodynamic equations
cites work: On the global well-posedness of \(N\)-dimensional generalized MHD system in anisotropic spaces
cites work: On the three-dimensional magnetohydrodynamics system in scaling-invariant spaces
cites work: Regularity criteria of the three-dimensional MHD system involving one velocity and one vorticity component
cites work: Regularity criteria of the 4D Navier-Stokes equations involving two velocity field components
cites work: Horizontal Biot-Savart law in general dimension and an application to the 4D magneto-hydrodynamics.
cites work: Regularity criteria for the generalized viscous MHD equations
cites work: On the regularity of the solutions of the Navier–Stokes equations via one velocity component
Wikidata QID: Q128667354
DOI: 10.4171/RMI/1034
description: scientific article; zbMATH DE number 7023569