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

Xiaoping  Zhai; Yongsheng Li; Wei Yan

doi:10.3934/cpaa.2015.14.1865

Article Contents

2015, Volume 14, Issue 5: 1865-1884. Doi: 10.3934/cpaa.2015.14.1865

This issue Previous Article Approximation schemes for non-linear second order equations on the Heisenberg group Next Article Maximal functions of multipliers on compact manifolds without boundary

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

1.
School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640
2.
Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640
3.
College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007

Received: August 31, 2014

Revised: January 31, 2015

Published: August 2015

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Abstract

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 $(u(t),b(t))\in C^{\infty}(\mathbb{R}^3)\times C^{\infty}(\mathbb{R}^3)$.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 76D03, 76W05.

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