Structural Decomposition of Thermo-elastic Semigroups with Rotational Forces

Abstract

B.C.~such as: hinged mechanical/Neumann thermal B.C., and the most challenging case of all, the so-called case of free B.C. In all cases, the original thermo-elastic s.c.~semigroup of contractions admits a structural decomposition, for all positive times, as the sum of a “non-compact semigroup” and a compact component. In all cases, save (at present) the case of free B.C., the “non-compact semigroup” component is actually a s.c. uniformly (exponentially) stable group, based only on the mechanical variables: as a consequence, a precise uniform (exponential) stability result of the original thermo-elastic semigroup is then obtained. For the free B.C. case, the “non-compact” semigroup corresponds to a simpler problem with uncoupled elastic equation and shear force B.C. The stated structural decomposition requires, for its proof, sharp/optimal regularity theory of the associated elastic Kirchoff equation; including two new such results as in [27,28] for hinged/Neumann and for free B.C., respectively. The structural decomposition results of this paper for models that account for rotational forces are at striking contrast with the property of analyticity of the thermo-elastic semigroup, which characterizes, instead, models which do not account for rotational forces. Implications on exact controllability are noted.