American Institute of Mathematical Sciences

Advanced
  • Previous Article
    POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds
  • DCDS-S Home
  • This Issue
  • Next Article
    Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings
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

[1]

Simone Creo, Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. Approximation of a nonlinear fractal energy functional on varying Hilbert spaces. Communications on Pure & Applied Analysis, 2018, 17 (2) : 647-669. doi: 10.3934/cpaa.2018035
[2]

Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103
[3]

Sébastien Gadat, Laurent Miclo. Spectral decompositions and $\mathbb{L}^2$-operator norms of toy hypocoercive semi-groups. Kinetic & Related Models, 2013, 6 (2) : 317-372. doi: 10.3934/krm.2013.6.317
[4]

Tasnim Fatima, Ekeoma Ijioma, Toshiyuki Ogawa, Adrian Muntean. Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers. Networks & Heterogeneous Media, 2014, 9 (4) : 709-737. doi: 10.3934/nhm.2014.9.709
[5]

Adeolu Taiwo, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2733-2759. doi: 10.3934/jimo.2020092
[6]

L. Yu. Glebsky and E. I. Gordon. On approximation of locally compact groups by finite algebraic systems. Electronic Research Announcements, 2004, 10: 21-28.

[7]

Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005
[8]

Martha Garlick, James Powell, David Eyre, Thomas Robbins. Mathematically modeling PCR: An asymptotic approximation with potential for optimization. Mathematical Biosciences & Engineering, 2010, 7 (2) : 363-384. doi: 10.3934/mbe.2010.7.363
[9]

Liren Lin, I-Liang Chern. A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1119-1128. doi: 10.3934/dcdsb.2014.19.1119
[10]

Eduardo Casas, Mariano Mateos, Arnd Rösch. Finite element approximation of sparse parabolic control problems. Mathematical Control & Related Fields, 2017, 7 (3) : 393-417. doi: 10.3934/mcrf.2017014
[11]

Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic & Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027
[12]

Shinji Nakaoka, Hisashi Inaba. Demographic modeling of transient amplifying cell population growth. Mathematical Biosciences & Engineering, 2014, 11 (2) : 363-384. doi: 10.3934/mbe.2014.11.363
[13]

Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko. On a class of model Hilbert spaces. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5067-5088. doi: 10.3934/dcds.2013.33.5067
[14]

Kaitlyn (Voccola) Muller. A reproducing kernel Hilbert space framework for inverse scattering problems within the Born approximation. Inverse Problems & Imaging, 2019, 13 (6) : 1327-1348. doi: 10.3934/ipi.2019058
[15]

Virginia Agostiniani. Second order approximations of quasistatic evolution problems in finite dimension. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1125-1167. doi: 10.3934/dcds.2012.32.1125
[16]

Irene Benedetti, Nguyen Van Loi, Valentina Taddei. An approximation solvability method for nonlocal semilinear differential problems in Banach spaces. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 2977-2998. doi: 10.3934/dcds.2017128
[17]

Aviv Gibali, Dang Thi Mai, Nguyen The Vinh. A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications. Journal of Industrial & Management Optimization, 2019, 15 (2) : 963-984. doi: 10.3934/jimo.2018080
[18]

Mou-Hsiung Chang, Tao Pang, Moustapha Pemy. Finite difference approximation for stochastic optimal stopping problems with delays. Journal of Industrial & Management Optimization, 2008, 4 (2) : 227-246. doi: 10.3934/jimo.2008.4.227
[19]

Zhong-Ci Shi, Xuejun Xu, Zhimin Zhang. The patch recovery for finite element approximation of elasticity problems under quadrilateral meshes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 163-182. doi: 10.3934/dcdsb.2008.9.163
[20]

Zhiming Li, Yujun Zhu. Entropies of commuting transformations on Hilbert spaces. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5795-5814. doi: 10.3934/dcds.2020246

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (163)
  • HTML views (459)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences