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Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces

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  • In this paper, we consider the global well-posedness of the incompressible magnetohydrodynamic equations with initial data $(u_0,b_0)$ in the critical Besov space $\dot{B}_{2,1}^{1/2}(\mathbb{R}^3)\times \dot{B}_{2,1}^{1/2}(\mathbb{R}^3)$. Compared with [30], making full use of the algebraical structure of the equations, we relax the smallness condition in the third component of the initial velocity field and magnetic field. More precisely, we prove that there exist two positive constants $\varepsilon_0$ and $C_0$ such that if \begin{eqnarray} (\|u_0^h\|_{\dot{B}_{2,1}^{1/2}} +\|b_0^h\|_{\dot{B}_{2,1}^{1/2}}) \exp\{C_0(\frac{1}{\mu}+\frac{1}{\nu})^3 (\|u_0^3\|_{\dot{B}_{2,1}^{1/2}} +\|b_0^3\|_{\dot{B}_{2,1}^{1/2}})^2\} \le \varepsilon_0\mu\nu, \end{eqnarray} then the 3-D incompressible magnetohydrodynamic system has a unique global solution $(u,b)\in C([0,+\infty);\dot{B}_{2,1}^{1/2})\cap L^1((0,+\infty);\dot{B}_{2,1}^{5/2})\times C([0,+\infty);\dot{B}_{2,1}^{1/2})\cap L^1((0,+\infty);\dot{B}_{2,1}^{5/2}).$ Finally, we analyze the long behavior of the solution and get some decay estimates which imply that for any $t>0$ the solution $(u(t),b(t))\in C^{\infty}(\mathbb{R}^3)\times C^{\infty}(\mathbb{R}^3)$.
    Mathematics Subject Classification: Primary: 35Q53; Secondary: 76D03, 76W05.

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