GLOBAL WELL-POSEDNESS OF THE 2D MHD EQUATIONS OF DAMPED WAVE TYPE IN SOBOLEV SPACE\ast

The magnetohydrodynamic system of damped wave type (abbreviated as MHD-wave system) is formally derived from Maxwell's equations of electromagnetism by keeping the usually ignored small term involving the product of permittivity and magnetic permeability. When this term is ignored in the context of nonrelativistic charged fluid, one obtains the standard MHD system. This extra term in the MHD-wave system assumes the form-\partialttb with -> 0 being a small constant and b the magnetic field. Mathematically this term makes the global well-posedness problem much more challenging than the corresponding MHD system. Even the global existence and regularity problem for the 2D MHD-wave system appears to be open. This paper solves the global well-posedness problem in a critical Sobolev setting when -and the size of the initial data satisfy a suitable constraint. In addition, the solution of the MHD-wave system is shown to converge to that of the corresponding MHD system with an explicit rate. The energy method does not work here, and this paper presents a new approach.