Mathematical Study of the Small Oscillations of a Spherical Layer of Viscous Fluid about a Rigid Spherical Core in the Gravitational Field

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• 作者：Doretta Vivona (1)
  Pierre Capodanno (2)
  
  1. Dipartimento di Scienze di Base ed Applicate per l鈥橧ngegneria ; 鈥淪apienza鈥?University of Rome ; via A.Scarpa n.16 ; 00161 ; Rome ; Italy
  2. University of Franche-Compt茅 ; Besan莽on ; France
  
• 关键词：Primary 76D05 ; 76M30 ; 35P99 ; Secondary 20M99 ; Small oscillations ; viscous fluid ; spherical layer ; gravitational field ; spectral and evolution problems ; semigroups
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：12
• 期：1
• 页码：245-262
• 全文大小：463 KB
• 参考文献：1. Askerov N.G., Krein S.G., Laptev G.I.: A class of non selfadjoint boundary value problems. Dokl. Akad. Nauk. SSSR. 155(3), 499鈥?02 (1964)
  2. Askerov N.G., Krein S.G., Laptev G.I.: Oscillations of a viscous liquid and the associated operatorial equations. Dokl. Akad. Nauk. SSSR. 2, 115鈥?24 (1968)
  3. Capodanno, P.: Piccole oscillazioni di un liquido viscoso pesante in un recipiente aperto, Lecture in 鈥淪apienza鈥?University of Rome, Facolt谩 di Ingegneria, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate (2001)
  4. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2, Springer, Berlin (2000)
  5. Hobson, E.W.: The theory of spherical and ellipso茂dal harmonics, Chelsea Publishing Company, New York (1965)
  6. Kopachevkii, N.D., Krein, S.G.: Operator approach in linear problems of hydrodynamics, vol. 2. Birkh盲user (2003)
  7. Lions, J.L.: Equations differentielles op茅rationelles et probl茅mes aux limites. Springer, Berlin (2000)
  8. Mikhlin, S.G.: Mathematical Physics, an advanced course. North Holland Publishing Company, Amsterdam (1970)
  9. Hubert Sanchez, J., Palencia Sanchez, E.: Vibration and coupling of continuous systems. Asymptotic methods. Springer, Berlin (1989)
  10. Wavre R.: Essai sur les petites vibrations des astres fluides. Commentarii Mathematici Helvetici 4, 74鈥?6 (1932) CrossRef
  11. Wavre, R.: Figures plan茅taires et g茅od茅sie. Gauthier-Villars, Paris (1932)
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1660-5454

文摘

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 using his method of the uniform process. In this paper, the authors consider the case of a layer of viscous fluid. Using the methods of functional analysis, they obtain from the variational form of the equations of motions, an operatorial equation in a Hilbert space. They reduce the problem of the small oscillations to the study of an operator pencil and, so, they can determine the spectrum of the problem and specify the set of eigenvalues according to the coefficient of viscosity. They also give results concerning the solutions of the evolution problem by means of the theory of the semi-group.