ABSTRACT

In this talk, I first recall some mathematical results in the study of classical Prandtl boundary layer theory. And then, I will focus on the related well-posedness and convergence theories for the characteristic MHD boundary layer. Under the assumption that the initial tangential magnetic field is not zero, we establish the local-in-time existence and uniqueness of solution to the nonlinear MHD boundary layer equations. Moreover, based on the multi-scale expansions, we justify the vanishing viscosity and magnetic diffusion limit process in $L^\infty$ sense by weighted energy estimates in Sobolev spaces. Compared with the classical Prandtl theory for which the monotonicity condition of the tangential velocity plays a crucial role, this monotonicity condition is not needed for MHD boundary layer. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics.