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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 $$\left( {\omega _\theta ,\nabla \left( {u_\theta e_\theta } \right)} \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$$ .
Acta Mathematicae Applicatae Sinica – Springer Journals
Published: Apr 10, 2013
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