Two-dimensional Boussinesq equation in a disc and anisotropic Sobolev spaces

Vladimir Varlamov

doi:10.1016/j.crme.2007.08.008

Boussinesq equation, elasticity, beams, plates

Two-dimensional Boussinesq equation in a disc and anisotropic Sobolev spaces
[L'équation de Boussinesq à deux dimensions dans un disque et les espaces de Sobolev anisotropes]

Présenté par : Presented by

Vladimir Varlamov ¹

¹ Department of Mathematics, University of Texas–Pan American, Edinburg, TX 78539-2999, USA

Comptes Rendus. Mécanique, Volume 335 (2007) no. 9-10, pp. 548-558.

Résumé
Abstract

L'équation de Boussinesq amortie à deux dimensions avec terme de forçage est considérée dans un disque unité. Elle gouverne les petites oscillations non linéaires, forcées d'une membrane élastique fine en présence de viscosité. La méthode de développement en fonctions propres est utilisée pour la construction des solutions globales en temps de problème mixte considéré. Des espaces anisotropes de Sobolev spécialement conçus, sont introduits pour démontrer l'effet de régularité non linéaire dans la coordonnée angulaire. L'existence et l'unicité dans ces espaces sont prouvées sur la base de cette construction.

The two-dimensional damped Boussinesq equation with a forcing term is considered in a unit disc. It governs forced, small, nonlinear oscillations of a thin elastic membrane in the presence of viscosity. The eigenfunction expansion method is used for constructing global-in-time solutions of the initial-boundary-value problem in question. Specially designed anisotropic Sobolev spaces are introduced in order to reflect the effect of nonlinear smoothing in the angular coordinate. Existence and uniqueness in these spaces are proved on the basis of the construction.

Reçu le : 2006-11-14
Accepté le : 2006-12-06
Publié le : 2007-09-01

DOI : 10.1016/j.crme.2007.08.008

Keywords: Vibrations, Boussinesq equation, Disc, Anisotropic Sobolev spaces
Mot clés : Vibrations, Équation de Boussinesq, Disque, Espaces de Sobolev anisotropes

Affiliations des auteurs :

Vladimir Varlamov ¹

¹ Department of Mathematics, University of Texas–Pan American, Edinburg, TX 78539-2999, USA

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Vladimir Varlamov. Two-dimensional Boussinesq equation in a disc and anisotropic Sobolev spaces. Comptes Rendus. Mécanique, Volume 335 (2007) no. 9-10, pp. 548-558. doi : 10.1016/j.crme.2007.08.008. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.08.008/

Bibliographie
Cité par

[1] J. Boussinesq Étude nouvelle sur l'équilibre et le mouvement des corps solides élastiques dont certaines dimensions sont très petites par rapport à d'autres. Premier mémoire : des tiges. Second mémoire : des plaques planes, J. Math. Pures Appl., Ser. II, Volume 16 (1871), pp. 125-274

[2] J. Boussineq Complément à une étude de 1871 sur la théorie de l'équilibre et du mouvement des solides élastiques…, J. Math. Pures Appl., Ser. III, Volume 5 (1879), pp. 163-194 (and 329–344)

[3] J. Boussinesq Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal…, J. Math. Pures Appl., Ser. II, Volume 17 (1872), pp. 55-108

[4] J. Bona; R. Sachs Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., Volume 118 (1988), pp. 15-29

[5] P. Deift; C. Tomei; E. Trubowitz Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., Volume 35 (1982), pp. 567-628

[6] R. Hirota Solutions of the classical Boussinesq equation and the spherical Boussinesq equation: the Wronskian technique, J. Phys. Soc. Japan, Volume 55 (1986), pp. 2137-2150

[7] F. Linares; M. Scialom Asymptotic behavior of solutions of a generalized Boussinesq-type equation, Nonlinear Anal., Volume 25 (1995), pp. 1147-1158

[8] Y. Liu Instability and blow up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal., Volume 26 (1995), pp. 1527-1546

[9] A.K. Pani; H. Saranga Finite element Galerkin method for the “good” Boussinesq equation, Nonlinear Anal., Volume 29 (1997) no. 8, pp. 937-956

[10] R.L. Pego; M.I. Weinstein Eigenvalues and instabilities of solitary waves, Phil. Trans. Roy. Soc. London A, Volume 340 (1992), pp. 47-94

[11] M. Tsutsumi; T. Matahashi On the Cauchy problem for the Boussinesq-type equation, Math. Japonica, Volume 36 (1991), pp. 371-379

[12] R.S. Johnson A two-dimensional Boussinesq equation for water waves and some of its solutions, J. Fluid Mech., Volume 323 (1996), pp. 65-78

[13] D.G. Akmel Global existence and decay for solutions to the “bad” Boussinesq equation in two space dimensions, Appl. Anal., Volume 83 (2004) no. 1, pp. 17-36

[14] J. Bona; L. Luo More results on the decay of solutions to nonlinear dispersive wave equations, Discrete & Continuous Dynamical Systems, Volume 1 (1995), pp. 151-193

[15] R.B. Guenther; J.W. Lee Partial Differential Equations of Mathematical Physics and Integral Equations, Prentice Hall, New Jersey, 1988

[16] J.E. Lagnese; G. Leugering; E.J.P.G. Schmidt Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser, Boston, 1994

[17] V. Varlamov On the initial-boundary value problem for the damped Boussinesq equation, Discrete & Continuous Dynamical Systems, Volume 4 (1998) no. 3, pp. 431-444

[18] V. Varlamov On the spatially two-dimensional Boussinesq equation in a circular domain, Nonlinear Anal., Volume 46 (2001), pp. 699-725

[19] V. Varlamov Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation, Discrete & Continuous Dynamical Systems, Volume 7 (2001) no. 4, pp. 675-702

[20] S.M. Nikolskii Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Berlin, New York, 1975

[21] V. Varlamov Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl., Volume 306 (2005), pp. 413-424

[22] V. Varlamov Special functions arising in the study of semi-linear equations in circular domains, J. Comp. Appl. Math., Volume 202 (2007), pp. 105-121

[23] G.N. Watson A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, London, 1966

[24] G. Avalos; I. Lasiecka Mechanical and thermal null controllability of thermoelastic plates and singularity of the associated minimal energy function, Control and Cybernetics, Volume 32 (2003) no. 3, pp. 473-490

[25] L.D. Landau; E.M. Lifshitz Theory of Elasticity, Pergamon, New York, 1964

[26] G. Tolstov Fourier Series, Dover Publ., New York, 1962

[27] F.W.J. Olver Introduction to Asymptotics and Special Functions, Acad. Press, New York, San Fransisco, London, 1974

[28] C.H. Jenkins; J.W. Leonard Nonlinear dynamic response of membranes: state of the art, Appl. Mech. Rev., Volume 44 (1991) no. 7, pp. 319-328

[29] V. Varlamov; A. Balogh Forced nonlinear oscillations of elastic membranes, Nonlinear Anal., Volume 7 (2006), pp. 1005-1028

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