A direct proof of the Caffarelli-Kohn-Nirenberg theorem

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

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Abstract

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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 $limsu{p}_{R\to 0⁺}1/R{\int }_{{Q}_{R}\left(x₀,t₀\right)}|curlu×u/|u||²dxdt\le {\epsilon }_{*}$ for a sufficiently small ${\epsilon }_{*}>0$.

How to cite

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Jörg 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 -

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