Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations

Kyungkeun Kang, Jihoon Lee

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

We present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three: a suitable weak solution is regular near an interior point z if the scaled Lx, t p, q-norm of the velocity with 1 ≤ 3 / p + 2 / q ≤ 2, 1 ≤ q ≤ ∞ is sufficiently small near z and if the scaled Lx, t l, m-norm of the magnetic field with 1 ≤ 3 / l + 2 / m ≤ 2, 1 ≤ m ≤ ∞ is bounded near z. Similar results are also obtained for the vorticity and for the gradient of the vorticity. Furthermore, with the aid of the regularity criteria, we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations.

Original languageEnglish
Pages (from-to)2310-2330
Number of pages21
JournalJournal of Differential Equations
Volume247
Issue number8
DOIs
Publication statusPublished - 2009 Oct 15

Fingerprint

Suitable Weak Solutions
Regularity Criterion
Magnetohydrodynamic Equations
Magnetohydrodynamics
Vorticity
Interior
Norm
Interior Point
Magnetic fields
Regularity Conditions
Weak Solution
Three-dimension
Magnetic Field
Gradient

All Science Journal Classification (ASJC) codes

  • Analysis

Cite this

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abstract = "We present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three: a suitable weak solution is regular near an interior point z if the scaled Lx, t p, q-norm of the velocity with 1 ≤ 3 / p + 2 / q ≤ 2, 1 ≤ q ≤ ∞ is sufficiently small near z and if the scaled Lx, t l, m-norm of the magnetic field with 1 ≤ 3 / l + 2 / m ≤ 2, 1 ≤ m ≤ ∞ is bounded near z. Similar results are also obtained for the vorticity and for the gradient of the vorticity. Furthermore, with the aid of the regularity criteria, we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations.",
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Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations. / Kang, Kyungkeun; Lee, Jihoon.

In: Journal of Differential Equations, Vol. 247, No. 8, 15.10.2009, p. 2310-2330.

Research output: Contribution to journalArticle

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AB - We present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three: a suitable weak solution is regular near an interior point z if the scaled Lx, t p, q-norm of the velocity with 1 ≤ 3 / p + 2 / q ≤ 2, 1 ≤ q ≤ ∞ is sufficiently small near z and if the scaled Lx, t l, m-norm of the magnetic field with 1 ≤ 3 / l + 2 / m ≤ 2, 1 ≤ m ≤ ∞ is bounded near z. Similar results are also obtained for the vorticity and for the gradient of the vorticity. Furthermore, with the aid of the regularity criteria, we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations.

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