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.

In this work, we investigated the local existence of the solutions to the 2D magnetohy-drodynamic (MHD) boundary layer equations on the half plane by energy methods in weighted Sobolev space. Compared to the existence of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays an important role, we used the initial tangential magnetic field with a lower bound $\delta >0$ instead of the monotonicity condition of the tangential velocity.

The objective of this paper is to investigate the local well-posedness of analytic solutions to the three-dimensional (3D) magnetohydrodynamic (MHD) boundary layer equations without structural assumptions. Specifically, the general initial data are required to be real-analytic in the tangential variables $(x,y)$ and satisfy Sobolev regularity in the normal variable $z$. We first adopt a variable transformation involving $\varphi (z)$ to homogenize the boundary conditions and eliminate resulting high-order terms. Additionally, we employ delicate energy estimates combined with Gauss weight functions to control linearly growth terms.