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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 no163, 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 no163, 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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