Local kinetic energy and singularities of the incompressible Navier–Stokes equations

Hi Jun Choe; Minsuk Yang

doi:10.1016/j.jde.2017.09.036

Local kinetic energy and singularities of the incompressible Navier–Stokes equations

Hi Jun Choe, Minsuk Yang

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to L^∞(0,T;L^3,w(R³)) where L^3,w(R³) denotes the standard weak Lebesgue space.

Original language	English
Pages (from-to)	1171-1191
Number of pages	21
Journal	Journal of Differential Equations
Volume	264
Issue number	2
DOIs	https://doi.org/10.1016/j.jde.2017.09.036
Publication status	Published - 2018 Jan 15

Bibliographical note

Funding Information:
Hi Jun Choe has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2015R1A5A1009350 and No. 2015R1A2A01002708 ). Minsuk Yang has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1C1B2015731 ).

Publisher Copyright:
© 2017 Elsevier Inc.

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

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10.1016/j.jde.2017.09.036

Cite this

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title = "Local kinetic energy and singularities of the incompressible Navier–Stokes equations",

abstract = "We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse H{\"o}lder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to L∞(0,T;L3,w(R3)) where L3,w(R3) denotes the standard weak Lebesgue space.",

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AB - We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to L∞(0,T;L3,w(R3)) where L3,w(R3) denotes the standard weak Lebesgue space.

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Local kinetic energy and singularities of the incompressible Navier–Stokes equations

Abstract

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