Mathematical study of the small oscillations of a spherical layer of viscoelastic fluid about a rigid spherical core in the gravitational field

The problem of the small oscillations of a spherical layer of an inviscid fluid about a rigid spherical body in the gravitational field has been studied by Laplace in the case of a fluid layer of small depth. His results have been rediscovered by R. Wavre by using his method of the uniform process. The second author and his collaborators have studied the case of a layer of viscous fluid by means of the methods of the functional analysis. In this paper, we consider the case of a layer of viscoelastic fluid that obeys to the simpler Oldroyd's law. Using the classical methods for the calculation of the potential and the methods of the functional analysis, we obtain from the variational form of the equations of the motion, an operatorial equation in a suitable Hilbert space. We reduce the problem of the small oscillations to the study of an operator pencil and so, we can precise the location of the spectrum and prove the existence of three sets of real eigenvalues. We give a theorem of existence and unicity of the solution of the associated evolution problem by means of the semi - groups theory.