On a magnetohydrodynamic problem of Euler type

Abstract

The magnetohydrodynamic (or MHD) equations of an incompressible and homogeneous plasma are considered in the limit of vanishing viscosity and resistivity, the plasma spreading over a smoothly bounded domain G of R N ( N⩾2). The problem is posed and solved as an equation of evolution in a space of divergence-free vector fields parallel to the boundary of G. Main results are existence, uniqueness, and regularity of maximal solutions, and continuous dependence on forcing terms and initial data. The proof of existence is based on Galerkin's method and a priori estimates; the continuity result is obtained by use of a regularization procedure—The approach is inspired by the work of R. Temam and T. Kato and C. Y. Lai on the Euler equations of ideal fluid flow.