Well-Posedness for the Three Dimension Magnetohydrodynamic System in the Anisotropic Besov Spaces
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Well-Posedness for the Three Dimension Magnetohydrodynamic System in the Anisotropic Besov Spaces Xiaoping Zhai1 · Yongsheng Li1 · Wei Yan2
Received: 9 December 2014 / Accepted: 10 June 2015 / Published online: 18 June 2015 © Springer Science+Business Media Dordrecht 2015
Abstract In this paper, we consider the global well-posedness of the incompressible magnetohydrodynamic system with anisotropic Besov initial data in the critical Besov spaces. Under the nonlinear smallness condition on the critical anisotropic Besov norm of the horizontal components of the initial velocity field and magnetic field, we get the system has a unique global solution. Our result allows to construct global solutions for a class of highly oscillating initial data of Cannone’s type. Keywords Well-posedness · MHD system · Anisotropic Besov spaces Mathematics Subject Classification 35Q35 · 76D03
1 Introduction and the Main Result In this paper, we consider the following incompressible magnetohydrodynamic system: ∂t u − μu + u · ∇u + ∇P − b · ∇b = 0,
(1.1)
∂t b − νb + u · ∇b − b · ∇u = 0,
(1.2)
This work is supported by NSFC under grant number 11171116 and 11401180, and by the Fundamental Research Funds for the Central Universities of China under the grant number 2012ZZ0072.
B X. Zhai
[email protected] Y. Li [email protected] W. Yan [email protected]
1
School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, P.R. China
2
College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, P.R. China
2
X. Zhai et al.
∇ · u = ∇ · b = 0, u(x, 0) = u0 ,
b(x, 0) = b0 ,
(1.3) (1.4)
where u = (u1 , u2 , u3 ) is the velocity field, b = (b1 , b2 , b3 ) is the magnetic field, P (x, t) is the scalar pressure, u0 = (uh0 , u30 ) and b0 = (b0h , b03 ) are the initial velocity field and the initial magnetic field respectively, μ > 0 is the viscosity coefficient and ν > 0 is the magnetic diffusive coefficient. The MHD system is a well-known model which governs the dynamics of the velocity and magnetic fields in electrically conducting fluids such as plasmas, liquid metals, and salt water, etc. In the case when the magnetic field b(x, t) is identically equal to zero, that is, in the case of the incompressible Navier-Stokes equations, the well-posedness in classical function spaces has been studied by many authors. For example, Fujita and Kato [10] proved both the global well-posedness for small initial data and the local well-posedness for large initial data in the Sobolev space H s (Rn ) for s ≥ n2 − 1. Similar results have been established in Ln (Rn ) −1+ n
by Kato [11], in the Besov space B˙ p,∞ p (Rn ), 1 ≤ p < ∞ with n < p < ∞ by Cannone [4] and Planchon [19], and in the larger space BMO−1 by Koch and Tataru [12] (see also [14, Chap. 16]). For the MHD system, the situation is more complicated due to the coupling effect between the velocity u(x, t) and the magnetic field b(x, t). It has been a long-standing open problem that whether or not classical solutions of (1.
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