Global regularity to the 3D incompressible MHD equations

In this paper, we prove the global regularity to the 3D nonhomogeneous incompressible magnetohydrodynamic equations. Let ϱ0ϱ0, m0m0 and H0H0 be the initial density, momentum and magnetic field, respectively. We establish a unique strong solution on R3×(0,T)R3×(0,T) for any 00μ>0 is sufficiently large, or ‖|m0|2/ϱ0‖L12+‖H0‖L22 or ‖∇u0‖L22+‖H0‖H12 is small enough. Moreover, if the given data are more regular and satisfy an additional compatibility condition for the existence of strong solution, then we show that the strong solution is indeed a classical one. Moreover, the weak–strong uniqueness of solutions is also obtained, which generalizes the result in [13] to the case of vacuum.