The well-posedness of the incompressible Magnetohydro Dynamic equations in the framework of Fourier-Herz space

作者：Li, Jingyue; Zheng, Xiaoxin^*

来源：Journal of Differential Equations, 2017, 263(6): 3419-3459.

DOI：10.1016/j.jde.2017.04.027

• Magnetohydro Dynamic equations
• Well-posedness
• Fourier-Herz space
• The blow-up criterion

In this paper, we study the well-posedness and the blow-up criterion of the mild solution for the 3D incompressible MHD equations in the framework of Fourier-Herz space involving highly oscillating function. First, we study the well-posedness of the incompressible MHD equations by establishing the smoothing effect in the mixed time space Fourier-Herz space, which include the local in time for large initial data as well as the global well-posedness for small initial data. Next, we prove the blow-up criterion, that is, if u is an element of L-T((r1) over tilde) F(B)over dot(p,q)(2-3/p+2/r1) and b is an element of (LTF)-F-(r2) over tilde(B)over dot(p,q)(2-3/p+2/r2) for 1 <= r(1), r(2) < infinity, the mild solution to the MHD equations can be extended beyond t = T. More importantly, we give a better blow-up criterion in which we require velocity field u(t) is an element of (LTF)-F-(r) over tilde(B)over dot(p,q)(2-3/p+2/r) only.

• 出版日期2017-9-15
• 单位中国工程物理研究院; 北京航空航天大学