On some recent results about inertial manifolds and kinematic dynamos

Abstract

The conditions imposed in the paper ['Inertial manifolds and completeness of eigenmodes for unsteady magnetic dynamos', Physica D {\bf 194} (2004) 297-319] on the fluid velocity to guarantee the existence of inertial manifolds for the kinematic dynamo problem are too demanding, in the sense that they imply that all the solutions tend exponentially to zero. The inertial manifolds are meaningful because they represent different decay rates, but the classical kinematic dynamos where the magnetic field is maintained or grows are not covered by this approach, at least until more refined estimates are found.