Regularity criteria of axisymmetric weak solutions to the 3D magnetohydrodynamic equations |
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Authors: | Bao-quan Yuan Feng-ping Li |
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Institution: | 1223. School of Mathematics and Information Science, Henan Polytechnic University, Henan, 454000, China
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Abstract: | In this paper, we study the regularity criteria for axisymmetric weak solutions to the MHD equations in ?3. 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 $\left( {\omega _\theta ,\nabla \left( {u_\theta e_\theta } \right)} \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$ . |
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Keywords: | regularity criteria axisymmetric solutions Besov space |
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