Nonlinear Stability in Fluids and Plasmas

Marsden, J. E. and T. S. Ratiu

Seminar on New Results in Nonlinear Partial Differential Equations, (A. J. Tromba, ed.), (1987), 101-134


Arnold's geometric methods are used to establish physically meaningful stability criteria for the Kelvin-Stuart cat's eye solution for two dimensional ideal fluids, vortex patches, reduced magnetohydrodynamics, and oceanographically interesting stratified shear flows.
This paper discuss some recent progress in the filed of stability of fluid and plasma equilibria. The objective is to drive explicit criteria which guarantee the nonlinear stability of specific equilibria. Most of the work described was done by H. Abarbanel, V. Arnold, R. Hazeltine, D. Holm, P. Morrison, M. Pulvirente, T. Ratiu, Y. Tang, Y.H. Wan, A. Weinstein and the author, although others have been involved in related work cited in the paper.