| Abstract: |
| This talk is concerned with a system of the 2D MHD equations with the kinematic dissipation given by the fractional operator and the magnetic diffusion by partial Laplacian. We are able to show that this system with any \alpha>0 always possesses a unique global smooth solution when the initial data is sufficiently smooth. In addition, we also give the optimal large-time decay rates. The second part is devoted to existence and regularity for a system of the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only directional hyperresistivity. More precisely, the equation of b_1(the horizontal component of the magnetic field) involves only vertical hyperdiffusion while the equation of b_2(the vertical component) has only horizontal hyperdiffusion. We prove that for derivative index great than 1 , this system always possesses a unique global-in-time classical solution when the initial data is sufficiently smooth. |
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