Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media

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October 2019, 12(6): 1709-1741. doi: 10.3934/dcdss.2019114

Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media

Christian Licht ^1,2,, and Thibaut Weller ^1,

1.	LMGC, Univ Montpellier, CNRS, Montpellier, France
2.	Dept. Maths, Mahidol University, Bangkok, Thailand

* Corresponding author: Christian Licht

Received December 2017 Revised March 2018 Published November 2018

We derive several models in Physics of continuous media using Trotter theory of convergence of semi-groups of operators acting on variable spaces.

Keywords: Asymptotic mathematical modeling, approximation of semi-groups in the sense of Trotter, Hilbert spaces of possible states with finite energy, transient problems, homogenization and reduction of dimension.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: Christian Licht, Thibaut Weller. Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1709-1741. doi: 10.3934/dcdss.2019114

References:

[1]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. Google Scholar
[2]	J. T. Beale, Eigenfunction expansions for objects floating in an open sea, Communications on Pure and Applied Mathematics, 30 (1977), 283-313. doi: 10.1002/cpa.3160300303. Google Scholar
[3]	D. Blanchard and G. A. Francfort, Asymptotic thermoelastic behavior of flat plates, Quart. Appl. Math., 45 (1987), 645-667. doi: 10.1090/qam/917015. Google Scholar
[4]	A. Bobrowski and M. Kimmel, An operator semigroup in mathematical genetics, SpringerBriefs in Applied Sciences and Technology, Mathematical Methods, Springer, 2015. doi: 10.1007/978-3-642-35958-3. Google Scholar
[5]	A. Bobrowski, Convergence of one-parameter operator semi-groups in models of mathematical biology and elsewhere, New Mathematical Monographs, 30, Cambridge University Press, 2016. doi: 10.1017/CBO9781316480663. Google Scholar
[6]	E. Bonetti, G. Bonfanti, C. Licht and R. Rossi, Dynamics of two linearly elastic bodies connected by a heavy thin soft viscoelastic layer, work in progress.Google Scholar
[7]	S. Brahim-Otsmane, G. A. Francfort and F. Murat, Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl., 71 (1992), 197-231. Google Scholar
[8]	H. Brezis, Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, Math. Studies, no. 5, North-Holland, Amsterdam, 1973. Google Scholar
[9]	P. G. Ciarlet, Mathematical elasticity, vol. II: theory of plates, North-Holland, Elsevier, 1997. Google Scholar
[10]	G. A. Francfort and P. Suquet, Homogenization and mechanical dissipation in thermo-viscoelasticity, Archive Rat. Mech. Anal., 96 (1986), 265-293. doi: 10.1007/BF00251909. Google Scholar
[11]	R. M. Garipov, On the linear theory of gravity waves, Archive Rat. Mech. Anal., 24 (1967), 352-362. doi: 10.1007/BF00253152. Google Scholar
[12]	P. Germain, Q. S. Nguyen and P. Suquet, Continuum thermodynamics, J. Appl. Mech., 50 (1983), 1010-1020. Google Scholar
[13]	B. Halphen and Q. S. Nguyen, Sur les matériaux standard généralisés, Journal de Mécanique, 14 (1975), 39-63. Google Scholar
[14]	O. Iosifescu, C. Licht and G. Michaille, Nonlinear boundary conditions in Kirchhoff-Love plate theory, J Elast, 96 (2009), 57-79. doi: 10.1007/s10659-009-9198-0. Google Scholar
[15]	T. Kato, Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, Proc. Japan Acad., 35 (1959), 467-468. doi: 10.3792/pja/1195524254. Google Scholar
[16]	C. Licht, Etude théorique et numérique de l'évolution d'un système fluide-flotteur, Thèse de docteur ingénieur, Nantes, 1980.Google Scholar
[17]	C. Licht, Etude de quelques modèles décrivant les vibrations d'une structure élastique dans la mer, Rapport de recherche de l'Ecole Nationale Supérieure des Techniques Avancées ENSTA n^o163, 1982.Google Scholar
[18]	C. Licht, Evolution d'un système fluide-flotteur, Journal de Mécanique Théorique et Appliquée, 1 (1982), 211-235. Google Scholar
[19]	C. Licht, Trois modèles décrivant les vibrations d'une structure dans la mer, C. R. Acad. Sci. Paris, Ser. I, 296 (1983), 341-344. Google Scholar
[20]	C. Licht, Comportement asymptotique d'une bande dissipative mince de faible rigidité, C. R. Acad. Sci. Paris, Ser. I, 317 (1993), 429-433. Google Scholar
[21]	C. Licht, Asymptotic behaviour of a thin dissipative layer, 2nd International Conference on Nonlinear Mechanics, Beijing, China, August 23-26, (1993), Ed. W.Z. Chien, Beijing University Press, 170-173.Google Scholar
[22]	C. Licht, Thin linearly viscoelastic Kelvin-Voigt plates, C. R. Mecanique, 341 (2013), 697-700. Google Scholar
[23]	C. Licht, A. Léger and F. Lebon, Dynamics of elastic bodies connected by a thin adhesive layer, Cinquièmes journées du GDR 'Étude de la propagation sonore en vue du contrôle non-destructif', Anglet, France, June 2-6, (2008), published in Ultrasonic wave propagation in non homogeneous media, Springer Proceedings in Physics 128, A. Léger and M. Deschamps Editors, Springer Verlag, 99-110.Google Scholar
[24]	C. Licht, A. Léger, S. Orankitjaroen and A. Ould Khaoua, Dynamics of elastic bodies connected by a thin soft viscoelastic layer, J. Math. Pures Appl., 99 (2013), 685-703. doi: 10.1016/j.matpur.2012.10.005. Google Scholar
[25]	C. Licht, S. Orankitjaroen, A. Ould Khaoua and T. Weller, Transient response of elastic bodies connected by a thin stiff viscoelastic layer with evanescent mass, C. R. Mecanique, 344 (2016), 736-743. Google Scholar
[26]	J. L. Lions, Réduction à des problèmes du type Cauchy-Kowaleska, Cours C.I.M.E., Cremonese, (1968), 269-280. Google Scholar
[27]	J. L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod, Gauthiers-Villars, Paris, 1968. Google Scholar
[28]	A. Raoult, Construction d'un modèle d'évolution de plaques avec terme d'inertie de rotation, Annali di Matematica Pura ed Applicata, 139 (1985), 361-400. doi: 10.1007/BF01766863. Google Scholar
[29]	H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math., 8 (1958), 887-919. doi: 10.2140/pjm.1958.8.887. Google Scholar
[30]	T. Weller, Etude des symétries et modèles de plaques en piézoélectricité linéarisée, Thèse, Montpellier, 2004.Google Scholar
[31]	T. Weller and C. Licht, Analyse asymptotique de plaques minces linéairement piézoélectriques, C. R. Acad. Sci. Paris, Ser. I, 335 (2002), 309-314. doi: 10.1016/S1631-073X(02)02457-3. Google Scholar
[32]	T. Weller and C. Licht, Asymptotic modeling of thin piezoelectric plates, Ann. Solid Struct. Mech., 1 (2010), 173-188. Google Scholar
[33]	V. V. Zhikov and S. E. Pastukhova, On the Trotter-Kato theorem in a variable space, Funct. Anal. Appl., 41 (2007), 264-270. doi: 10.1007/s10688-007-0024-9. Google Scholar

show all references

References:

[1]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. Google Scholar
[2]	J. T. Beale, Eigenfunction expansions for objects floating in an open sea, Communications on Pure and Applied Mathematics, 30 (1977), 283-313. doi: 10.1002/cpa.3160300303. Google Scholar
[3]	D. Blanchard and G. A. Francfort, Asymptotic thermoelastic behavior of flat plates, Quart. Appl. Math., 45 (1987), 645-667. doi: 10.1090/qam/917015. Google Scholar
[4]	A. Bobrowski and M. Kimmel, An operator semigroup in mathematical genetics, SpringerBriefs in Applied Sciences and Technology, Mathematical Methods, Springer, 2015. doi: 10.1007/978-3-642-35958-3. Google Scholar
[5]	A. Bobrowski, Convergence of one-parameter operator semi-groups in models of mathematical biology and elsewhere, New Mathematical Monographs, 30, Cambridge University Press, 2016. doi: 10.1017/CBO9781316480663. Google Scholar
[6]	E. Bonetti, G. Bonfanti, C. Licht and R. Rossi, Dynamics of two linearly elastic bodies connected by a heavy thin soft viscoelastic layer, work in progress.Google Scholar
[7]	S. Brahim-Otsmane, G. A. Francfort and F. Murat, Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl., 71 (1992), 197-231. Google Scholar
[8]	H. Brezis, Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, Math. Studies, no. 5, North-Holland, Amsterdam, 1973. Google Scholar
[9]	P. G. Ciarlet, Mathematical elasticity, vol. II: theory of plates, North-Holland, Elsevier, 1997. Google Scholar
[10]	G. A. Francfort and P. Suquet, Homogenization and mechanical dissipation in thermo-viscoelasticity, Archive Rat. Mech. Anal., 96 (1986), 265-293. doi: 10.1007/BF00251909. Google Scholar
[11]	R. M. Garipov, On the linear theory of gravity waves, Archive Rat. Mech. Anal., 24 (1967), 352-362. doi: 10.1007/BF00253152. Google Scholar
[12]	P. Germain, Q. S. Nguyen and P. Suquet, Continuum thermodynamics, J. Appl. Mech., 50 (1983), 1010-1020. Google Scholar
[13]	B. Halphen and Q. S. Nguyen, Sur les matériaux standard généralisés, Journal de Mécanique, 14 (1975), 39-63. Google Scholar
[14]	O. Iosifescu, C. Licht and G. Michaille, Nonlinear boundary conditions in Kirchhoff-Love plate theory, J Elast, 96 (2009), 57-79. doi: 10.1007/s10659-009-9198-0. Google Scholar
[15]	T. Kato, Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, Proc. Japan Acad., 35 (1959), 467-468. doi: 10.3792/pja/1195524254. Google Scholar
[16]	C. Licht, Etude théorique et numérique de l'évolution d'un système fluide-flotteur, Thèse de docteur ingénieur, Nantes, 1980.Google Scholar
[17]	C. Licht, Etude de quelques modèles décrivant les vibrations d'une structure élastique dans la mer, Rapport de recherche de l'Ecole Nationale Supérieure des Techniques Avancées ENSTA n^o163, 1982.Google Scholar
[18]	C. Licht, Evolution d'un système fluide-flotteur, Journal de Mécanique Théorique et Appliquée, 1 (1982), 211-235. Google Scholar
[19]	C. Licht, Trois modèles décrivant les vibrations d'une structure dans la mer, C. R. Acad. Sci. Paris, Ser. I, 296 (1983), 341-344. Google Scholar
[20]	C. Licht, Comportement asymptotique d'une bande dissipative mince de faible rigidité, C. R. Acad. Sci. Paris, Ser. I, 317 (1993), 429-433. Google Scholar
[21]	C. Licht, Asymptotic behaviour of a thin dissipative layer, 2nd International Conference on Nonlinear Mechanics, Beijing, China, August 23-26, (1993), Ed. W.Z. Chien, Beijing University Press, 170-173.Google Scholar
[22]	C. Licht, Thin linearly viscoelastic Kelvin-Voigt plates, C. R. Mecanique, 341 (2013), 697-700. Google Scholar
[23]	C. Licht, A. Léger and F. Lebon, Dynamics of elastic bodies connected by a thin adhesive layer, Cinquièmes journées du GDR 'Étude de la propagation sonore en vue du contrôle non-destructif', Anglet, France, June 2-6, (2008), published in Ultrasonic wave propagation in non homogeneous media, Springer Proceedings in Physics 128, A. Léger and M. Deschamps Editors, Springer Verlag, 99-110.Google Scholar
[24]	C. Licht, A. Léger, S. Orankitjaroen and A. Ould Khaoua, Dynamics of elastic bodies connected by a thin soft viscoelastic layer, J. Math. Pures Appl., 99 (2013), 685-703. doi: 10.1016/j.matpur.2012.10.005. Google Scholar
[25]	C. Licht, S. Orankitjaroen, A. Ould Khaoua and T. Weller, Transient response of elastic bodies connected by a thin stiff viscoelastic layer with evanescent mass, C. R. Mecanique, 344 (2016), 736-743. Google Scholar
[26]	J. L. Lions, Réduction à des problèmes du type Cauchy-Kowaleska, Cours C.I.M.E., Cremonese, (1968), 269-280. Google Scholar
[27]	J. L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod, Gauthiers-Villars, Paris, 1968. Google Scholar
[28]	A. Raoult, Construction d'un modèle d'évolution de plaques avec terme d'inertie de rotation, Annali di Matematica Pura ed Applicata, 139 (1985), 361-400. doi: 10.1007/BF01766863. Google Scholar
[29]	H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math., 8 (1958), 887-919. doi: 10.2140/pjm.1958.8.887. Google Scholar
[30]	T. Weller, Etude des symétries et modèles de plaques en piézoélectricité linéarisée, Thèse, Montpellier, 2004.Google Scholar
[31]	T. Weller and C. Licht, Analyse asymptotique de plaques minces linéairement piézoélectriques, C. R. Acad. Sci. Paris, Ser. I, 335 (2002), 309-314. doi: 10.1016/S1631-073X(02)02457-3. Google Scholar
[32]	T. Weller and C. Licht, Asymptotic modeling of thin piezoelectric plates, Ann. Solid Struct. Mech., 1 (2010), 173-188. Google Scholar
[33]	V. V. Zhikov and S. E. Pastukhova, On the Trotter-Kato theorem in a variable space, Funct. Anal. Appl., 41 (2007), 264-270. doi: 10.1007/s10688-007-0024-9. Google Scholar

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