# A direct proof of the Caffarelli-Kohn-Nirenberg theorem

Banach Center Publications (2008)

- Volume: 81, Issue: 1, page 533-552
- ISSN: 0137-6934

## Access Full Article

top## Abstract

top## How to cite

topJörg Wolf. "A direct proof of the Caffarelli-Kohn-Nirenberg theorem." Banach Center Publications 81.1 (2008): 533-552. <http://eudml.org/doc/282308>.

@article{JörgWolf2008,

abstract = {In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if
$lim sup_\{R→0⁺\} 1/R ∫_\{Q_R(x₀,t₀)\} |curl u × u/|u| |² dx dt ≤ ε_\{*\}$
for a sufficiently small $ε_\{*\} > 0$.},

author = {Jörg Wolf},

journal = {Banach Center Publications},

keywords = {Navier-Stokes equations; regularity of weak solutions; Caffarelli-Kohn-Nirenberg theorem},

language = {eng},

number = {1},

pages = {533-552},

title = {A direct proof of the Caffarelli-Kohn-Nirenberg theorem},

url = {http://eudml.org/doc/282308},

volume = {81},

year = {2008},

}

TY - JOUR

AU - Jörg Wolf

TI - A direct proof of the Caffarelli-Kohn-Nirenberg theorem

JO - Banach Center Publications

PY - 2008

VL - 81

IS - 1

SP - 533

EP - 552

AB - In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if
$lim sup_{R→0⁺} 1/R ∫_{Q_R(x₀,t₀)} |curl u × u/|u| |² dx dt ≤ ε_{*}$
for a sufficiently small $ε_{*} > 0$.

LA - eng

KW - Navier-Stokes equations; regularity of weak solutions; Caffarelli-Kohn-Nirenberg theorem

UR - http://eudml.org/doc/282308

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.