Abstract: In this talk, I shall introduce some recent results on global (or small) solutions of the incompressible magnetohydrodynamic(MHD) system with some kinds of dissipation terms in 2D，including the fractional Laplacian, anisotropic dissipation. These results established the global existence and uniqueness of smooth solutions when the initial data is close to an equilibrium state. Some of results presented rigorously confirm a nonlinear phenomenon observed in physical experiments that the magnetic field stabilizes electrically conducting fluids in fact, and some cases also gave the explicit large-time decay rates for various Sobolev norms of the solutions.
