Global well-posedness of the MHD boundary layer equation in the Sobolev Space

We study the two-dimensional MHD boundary layer equations. For small perturbation around a tangential background magnetic field, we obtain the global-in-time existence and uniqueness of solutions in Sobolev spaces. The proof relies on the novel combination of the well-explored cancellation mechanism and the idea of linearly-good unknowns, and in fact we use the former idea to deal with the top tangential derivatives and the latter one admitting fast decay rate to control lower-order derivatives.