On the global well-posedness to the 3-D Navier-Stokes-Maxwell system

Gaocheng  Yue; Chengkui Zhong

doi:10.3934/dcds.2016056

Article Contents

2016, Volume 36, Issue 10: 5817-5835. Doi: 10.3934/dcds.2016056

This issue Previous Article On the global well-posedness to the 3-D incompressible anisotropic magnetohydrodynamics equations Next Article

On the global well-posedness to the 3-D Navier-Stokes-Maxwell system

Gaocheng Yue^1, and
Chengkui Zhong^2,

1.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106
2.
Department of Mathematics, Nanjing University, Nanjing 210093

Received: August 31, 2015

Revised: February 29, 2016

Published: May 2017

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Abstract

The present paper is devoted to the well-posedness issue of solutions of a full system of the $3$-$D$ incompressible magnetohydrodynamic(MHD) equations. By means of Littlewood-Paley analysis we prove the global well-posedness of solutions in the Besov spaces $\dot{B}_{2,1}^\frac1{2}\times B_{2,1}^\frac3{2}\times B_{2,1}^\frac3{2}$ provided the norm of initial data is small enough in the sense that \begin{align*} \big(\|u_0^h\|_{\dot{B}_{2,1}^\frac1{2}} +\|E_0\|_{B_{2,1}^\frac{3}{2}}+\|B_0\|_{B_{2,1}^\frac{3}{2}}\big)\exp \Big\{\frac{C_0}{\nu^2}\|u_0^3\|_{\dot{B}_{2,1}^\frac1{2}}^2\Big\}\leq c_0, \end{align*} for some sufficiently small constant $c_0.$

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Mathematics Subject Classification: Primary: 35Q30, 76D03; Secondary: 76A05.

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