Asymptotic Stability for Equilibria of Nonlinear Semiflows with Applications to Rotating Viscoelastic Rods, Part I

Xu, C. Y. and J. E. Marsden

Top. Methods in Nonlin. Anal., 7, 271-297

Abstract:

This paper establishes abstract results, which extend those of Potier-Ferry and Sobolevskii, on global existence and stability of solutions to quasilinear equations near an equilibrium point whose spectrum lies in the strict left half plane. The result may be regarded as a version of the linearization principle for quasilinear systems in a context where the main difficulty is to show that near the equilibrium shocks are suppressed by small damping. In the second part to this work, applications are be made to the dynamics of rods undergoing uniform rotation and satisfying the formal stability criteria based on the energy-momentum method of Simo, Posbergh, and Marsden.

The stability of relative equilibria of dissipationless geometrically exact rods moving in space was analyzed by Simo, Posbergh, and Marsden [1990]. Applying the energy-momentum method, they obtained sufficient conditions for the formal stability of these relative equilibria. For these partial differential equations the theory only gives conditional stability since basic existence and uniqueness questions remain a difficulty due to the quasilinear nature of the equations and the associated problem of shock formation.

In this paper we prove that in the presence of dissipation (viscoelastic dissipation, for instance), formal stability also ensures the global existence of smooth solutions and nonlinear asymptotic dynamical stability for relative equilibria of geometrically exact rods (shells, etc.) moving in space. Since the system is free to rotate, the stability results are modulo appropriate rotations.

Early work in this direction was done by Browne [1978], who considered the problem of existence, uniqueness and stability for the quasilinear partial differential equations governing the motion of nonlinearly viscoelastic one-dimensional bodies.