DE Seminar: Rasika Mahawattege (UMBC)

Abstract: We consider a fluid-plate interaction model where the two dimensional plate is subject to viscoelastic (strong) damping. The strength of the Kelvin-Voigt damping is measured by a constant 0<rho<1. Coupling occurs at the interface between the two media, where each component evolves. In this paper, we apply “low" physically hinged boundary interface conditions, which involve the bending moment operator for the plate. We prove four main results: 

(i) analyticity, on the natural energy space, of the corresponding contraction semigroup (and of its adjoint); 

(ii) sharp location of the spectrum of its generator (and similarly of the adjoint generator), neither of which has compact resolvent, and in fact both of which have the point lambda=-1/rho in their respective continuous spectrum; 

(iii) both original generator and its adjoint have the origin lambda=0 as a common eigenvalue with a common, explicit, 1-dimensional eigenspace; 

(iv) The subspace of codimension 1 obtained by the original energy space by factoring out the common 1-dimensional eigenspace is invariant under the action of the (here restricted) semigroup (or of its adjoint), and on such subspace both original and adjoint semigroups are uniformly stable.