Nonlinear stability of ideal fluid equilibria

Conference · Fri Jan 01 04:00:00 EST 1988

OSTI ID:7182511

Holm, D D

The Lyapunov method for establishing stability is related to well- known energy principles for nondissipative dynamical systems. A development of the Lyapunov method for Hamiltonian systems due to Arnold establishes sufficient conditions for Lyapunov stability by using the energy plus other conserved quantities, together with second variations and convexity estimates. When treating the stability of ideal fluid dynamics within the Hamiltonian framework, a useful class of these conserved quantities consists of the Casimir functionals, which Poisson-commute with all functionals of the dynamical fluid variables. Such conserved quantities, when added to the energy, help to provide convexity estimates that bound the growth of perturbations. These convexity estimates, in turn, provide norms necessary for establishing Lyapunov stability under the nonlinear evolution. In contrast, the commonly used second variation or spectral stability arguments only prove linearized stability. As ideal fluid examples, in these lectures we discuss planar barotropic compressible fluid dynamics, the three-dimensional hydrostatic Boussinesq model, and a new set of shallow water equations with nonlinear dispersion due to Basdenkov, Morosov, and Pogutse(1985). Remarkably, all three of these samples have the same Hamiltonian structure and, thus, possess the same Casimir functionals upon which their stability analyses are based. We also treat stability of modified quasigeostrophic flow, a problem whose Hamiltonian structure and Casimirs closely resemble Arnold's original example. Finally, we discuss some aspects of conditional stability and the applicability of Arnold's development of the Lyapunov technique. 100 refs.

• Research Organization: Los Alamos National Lab., NM (USA)
• DOE Contract Number: W-7405-ENG-36
• OSTI ID: 7182511
• Report Number(s): LA-UR-88-1743; CONF-8806161-2; ON: DE88014477
• Country of Publication: United States
• Language: English

Related Subjects

640410* -- Fluid Physics-- General Fluid Dynamics
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
ALGORITHMS
CASIMIR EFFECT
DIFFERENTIAL EQUATIONS
EQUATIONS
EQUATIONS OF MOTION
FLUID FLOW
FLUID MECHANICS
HAMILTONIANS
HYDRODYNAMICS
LYAPUNOV METHOD
MATHEMATICAL LOGIC
MATHEMATICAL OPERATORS
MECHANICS
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM OPERATORS
STABILITY