The conditions for the nonlinear stability of plane and helical MHD flows
The stability of steady MHD flows of ideal incompressible fluid with uniform density is investigated. The authors consider only the flows which are invariant under translational, axial, rotational or helical symmetry. The sufficient conditions for nonlinear stability of such flows with respect to perturbations possessing the corresponding symmetry are obtained. These conditions are proved by constructing functionals with an absolute minima on particular steady solutions. Each of these functionals is a sum of kinetic energy, the integral of an arbitrary function of a Lagrangian coordinate, and an integral which is specific for the flows under consideration. By using the Lagrangian coordinate fields, one obtains a family of new stability definitions. According to these definitions, deviations of perturbed flows from unperturbed ones are measured by the integrals of the squared velocity and Lagrangian coordinate perturbations. The obtained stability conditions are extended to new types of flows.
scientific article; zbMATH DE number 1077503