Spatial decay estimates and continuous dependence on modelling for an equation from dynamo theory
Franca Franchi, Brian Straughan · Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences · 1994
Abstract This paper studies the improperly posed, backward in time problem, in addition to the well posed forward in time problem, for a non-symmetric partial differential equation which describes the behaviour of the toroidal part of the magnetic field in a dynamo problem. We first show that solutions in an unbounded cylinder decay exponentially in space provided that for the backward in time problem a Dirichlet integral is bounded and provided the prescribed velocity field satisfies particular bounds; for the forward in time problem several of these constraints are relaxed. It is next shown that the solution to the same problem on a bounded spatial domain depends Hölder continuously on changes in the prescribed velocity field.