Well-posedness of the kinematic dynamo problem

摘要

In the framework of magnetohydrodynamics, the generation of magnetic fields by the prescribed motion of a liquid conductor in a bounded region G??3 is described by the induction equation, a linear system of parabolic equations for the magnetic field components. Outside G, the solution matches continuously to some harmonic field that vanishes at spatial infinity. The kinematic dynamo problem seeks to identify those motions, which lead to nondecaying (in time) solutions of this evolution problem. In this paper, the existence problem of classical (decaying or not) solutions of the evolution problem is considered for the case that G is a ball and for sufficiently regular data. The existence proof is based on the poloidal/toroidal representation of solenoidal fields in spherical domains and on the construction of appropriate basis functions for a Galerkin procedure.