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.

拉普拉斯已经研究了在深度较小的流体层情况下，粘性流体的球形层围绕刚性球形体在重力场中的小振动问题。R. Wavre使用他的统一过程方法重新发现了他的结果。第二作者和他的合作者通过功能分析的方法研究了一层粘性流体的情况。在本文中，我们将考虑遵循简单的奥尔德罗伊德定律的粘弹性流体层的情况。使用经典的势能计算方法和泛函分析方法，我们从运动方程的变分形式中获得了合适的希尔伯特空间中的一个运算方程。我们通过研究算子铅笔来减少小振荡的问题，因此，我们可以精确地确定频谱的位置并证明三组真实特征值的存在。通过半群理论，给出了相关演化问题解的存在性和唯一性定理。