Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

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

Pages: 1061 - 1073, Volume 17, Issue 3, May 2012      doi:10.3934/dcdsb.2012.17.1061

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Keyan Wang - Department of Mathematics, Shanghai Finance University, Shanghai 201209, China (email)
Yi Du - Department of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China (email)

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.

Keywords:  Magnetohydrodynamic equations, large data, stability, global existence.
Mathematics Subject Classification:  Primary: 35Q30, 35B35; Secondary: 76D05,76E05.

Received: March 2011;      Revised: September 2011;      Available Online: January 2012.