Regularity criteria of axisymmetric weak solutions to the 3D magnetohydrodynamic equations

Regularity criteria of axisymmetric weak solutions to the 3D magnetohydrodynamic equations In this paper, we study the regularity criteria for axisymmetric weak solutions to the MHD equations in ℝ3. Let ω θ , J θ and u θ be the azimuthal component of ω, J and u in the cylindrical coordinates, respectively. Then the axisymmetric weak solution (u, b) is regular on (0, T) if (ω θ , J θ ) ∈ L q (0, T; L p ) or (ω θ , ▽(u θ e θ )) ∈ L q (0, T; L p ) with $$\tfrac{3} {p} + \tfrac{2} {q} \leqslant 2, \tfrac{3} {2} < p < \infty$$ . In the endpoint case, one needs conditions $$\left( {\omega _\theta ,J_\theta } \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$$ or $$\left( {\omega _\theta ,\nabla \left( {u_\theta e_\theta } \right)} \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Regularity criteria of axisymmetric weak solutions to the 3D magnetohydrodynamic equations

, Volume 29 (2) – Apr 10, 2013
14 pages

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Publisher
Springer Journals
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-013-0223-5
Publisher site
See Article on Publisher Site

Abstract

In this paper, we study the regularity criteria for axisymmetric weak solutions to the MHD equations in ℝ3. Let ω θ , J θ and u θ be the azimuthal component of ω, J and u in the cylindrical coordinates, respectively. Then the axisymmetric weak solution (u, b) is regular on (0, T) if (ω θ , J θ ) ∈ L q (0, T; L p ) or (ω θ , ▽(u θ e θ )) ∈ L q (0, T; L p ) with $$\tfrac{3} {p} + \tfrac{2} {q} \leqslant 2, \tfrac{3} {2} < p < \infty$$ . In the endpoint case, one needs conditions $$\left( {\omega _\theta ,J_\theta } \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$$ or $$\left( {\omega _\theta ,\nabla \left( {u_\theta e_\theta } \right)} \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$$ .

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Apr 10, 2013

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