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Nonlinear stability of fluid and plasma equilibria. (English) Zbl 0717.76051
Summary: The Lyapunov method for establishing stability has been used in a variety of fluid and plasma problems. For nondissipative systems, this stability method is related to well-known energy principles. A development of the Lyapunov method for Hamiltonian systems due to Arnold uses the energy plus other conserved quantities, together with second variations and convexity estimates, to establish stability. For Hamiltonian systems, a useful class of these conserved quantities consists of the Casimir functionals, which Poisson-commute with all functionals of the given dynamical variables. Such conserved quantities, when added to the energy, help to provide convexity estimates bounding the growth of perturbations. These estimates enable one to prove nonlinear stability, whereas the commonly used second variation or spectral arguments only prove linearized stability. When combined with recent advances in the Hamiltonian structure of fluid and plasma systems, this convexity method proves to be widely and easily applicable. This paper obtains new nonlinear stability criteria for equilibria for MHD, multifluid plasmas and the Maxwell-Vlasov equations in two and three dimensions. Related systems, such as multilayer quasigeostrophic flow, adiabatic flow and the Poisson-Vlasov equation are also treated. Other related systems, such as stratified flow and reduced magnetohydrodynamic equilibria are mentioned where appropriate, but are treated in detail in other publications.

MSC:
76E30 Nonlinear effects in hydrodynamic stability
76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
76A02 Foundations of fluid mechanics
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