A new kind of global solution for the MHD boundary layer system

Abstract

The MHD boundary layer describes the behavior of electrically conducting fluids (such as plasmas or liquid metals) in the presence of a magnetic field. By the Crocco inverse transformation, the MHD boundary layer system is transformed to a parabolic equation, which is called as the MHD boundary layer equation. Under the Oleinik assumption, the existence of the local analytic solution can be proved similar to the Prandtl boundary layer system. In order to overcome the difficulties arising from the degeneracy and the singularity of the MHD boundary layer equation, we used some innovative variable substitutions and introduce two new kinds of BV entropy solutions. By choosing a suitable test function, we were surprised to discover that the stability of entropy solutions can be proved independent of the boundary value condition. This novel finding provides a fresh perspective for re-examining relevant issues in the future, particularly regarding the potential significance of nonlinear boundary conditions historically imposed on MHD boundary layer equations.