Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows

An initial–boundary value problem is considered for the density-dependent incompressible viscous magnetohydrodynamic flow in a three-dimensional bounded domain. The homogeneous Dirichlet boundary condition is prescribed on the velocity, and the perfectly conducting wall condition is prescribed on the magnetic field. For the initial density away from vacuum, the existence and uniqueness are established for the local strong solution with large initial data as well as for the global strong solution with small initial data. Furthermore, the weak–strong uniqueness of solutions is also proved, which shows that the weak solution is equal to the strong solution with certain initial data.

► An initial–boundary value problem is studied for the magnetohydrodynamic fluids.
► The fluid is incompressible and density dependent.
► Local strong solution away from vacuum is obtained.
► Global strong solution with small initial data is obtained.
► The weak–strong uniqueness is proved.