Stability of the two dimensionalmagnetohydrodynamic flows in $\mathbb{R}^3$

Keyan Wang; Yi Du

doi:10.3934/dcdsb.2012.17.1061

Article Contents

2012, Volume 17, Issue 3: 1061-1073. Doi: 10.3934/dcdsb.2012.17.1061

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Stability of the two dimensional magnetohydrodynamic flows in $\mathbb{R}^3$

Keyan Wang^1, and
Yi Du^2,

1.
Department of Mathematics, Shanghai Finance University, Shanghai 201209
2.
Department of Mathematical Sciences, South China Normal University, Guangzhou, 510631

Received: March 2011

Revised: September 2011

Published: May 2012

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Abstract

We prove that two dimensional incompressible magnetohydrodynamic flows are stable in $\mathbb{R}^3$. As a corollary, we show the global existence of classical solutions to the three dimensional incompressible magnetohydrodynamic equations with small initial data. Furthermore, our smallness assumption of the perturbed initial data $(u_0, B_0)$ from that of the two dimensional case is only imposed on the scaling invariant quantity $\|u_0\|_{L^2}\|(\xi\cdot\nabla)u_0\|_{L^2} + \|B_0\|_{L^2}\|(\xi\cdot\nabla)B_0\|_{L^2}$ for one direction $\xi$, while $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} + \|B_0\|_{L^2}\|\nabla B_0\|_{L^2}$ may be arbitrarily large.

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Mathematics Subject Classification: Primary: 35Q30, 35B35; Secondary: 76D05,76E05.

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