Well-posedness of the MHD boundary layer equations with small initial data in Sobolev space

The purpose of this paper is to prove the well-posedness of the 2D magnetohydrodynamic (MHD) boundary layer equations for small initial data in Sobolev space of polynomial weight and low regularity. Our proofs are based on the paralinearization method and an abstract bootstrap argument. We first obtain the systems (3.3)–(3.6) by paralinearizing and symmetrizing the system (1.2). Then, we establish the estimates of the solution in horizontal direction and vertical direction, respectively. Finally, we prove the well-posedness of the 2D MHD boundary layer equations by an abstract bootstrap argument.