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Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media
1. | LMGC, Univ Montpellier, CNRS, Montpellier, France |
2. | Dept. Maths, Mahidol University, Bangkok, Thailand |
We derive several models in Physics of continuous media using Trotter theory of convergence of semi-groups of operators acting on variable spaces.
References:
[1] |
G. Allaire,
Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[8] |
H. Brezis,
Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, Math. Studies, no. 5, North-Holland, Amsterdam, 1973. |
[9] |
P. G. Ciarlet,
Mathematical elasticity, vol. II: theory of plates, North-Holland, Elsevier, 1997. |
[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. |
[11] |
R. M. Garipov,
On the linear theory of gravity waves, Archive Rat. Mech. Anal., 24 (1967), 352-362.
doi: 10.1007/BF00253152. |
[12] |
P. Germain, Q. S. Nguyen and P. Suquet,
Continuum thermodynamics, J. Appl. Mech., 50 (1983), 1010-1020.
|
[13] |
B. Halphen and Q. S. Nguyen,
Sur les matériaux standard généralisés, Journal de Mécanique, 14 (1975), 39-63.
|
[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. |
[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. |
[16] |
C. Licht,
Etude théorique et numérique de l'évolution d'un système fluide-flotteur, Thèse de docteur ingénieur, Nantes, 1980. |
[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. |
[18] |
C. Licht,
Evolution d'un système fluide-flotteur, Journal de Mécanique Théorique et Appliquée, 1 (1982), 211-235.
|
[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.
|
[20] |
C. Licht,
Comportement asymptotique d'une bande dissipative mince de faible rigidité, C. R. Acad. Sci. Paris, Ser. I, 317 (1993), 429-433.
|
[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. |
[22] |
C. Licht,
Thin linearly viscoelastic Kelvin-Voigt plates, C. R. Mecanique, 341 (2013), 697-700.
|
[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. |
[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. |
[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.
|
[26] |
J. L. Lions,
Réduction à des problèmes du type Cauchy-Kowaleska, Cours C.I.M.E., Cremonese, (1968), 269-280.
|
[27] |
J. L. Lions,
Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod, Gauthiers-Villars, Paris, 1968. |
[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. |
[29] |
H. F. Trotter,
Approximation of semi-groups of operators, Pacific J. Math., 8 (1958), 887-919.
doi: 10.2140/pjm.1958.8.887. |
[30] |
T. Weller,
Etude des symétries et modèles de plaques en piézoélectricité linéarisée, Thèse, Montpellier, 2004. |
[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. |
[32] |
T. Weller and C. Licht,
Asymptotic modeling of thin piezoelectric plates, Ann. Solid Struct. Mech., 1 (2010), 173-188.
|
[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. |
show all references
References:
[1] |
G. Allaire,
Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[8] |
H. Brezis,
Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, Math. Studies, no. 5, North-Holland, Amsterdam, 1973. |
[9] |
P. G. Ciarlet,
Mathematical elasticity, vol. II: theory of plates, North-Holland, Elsevier, 1997. |
[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. |
[11] |
R. M. Garipov,
On the linear theory of gravity waves, Archive Rat. Mech. Anal., 24 (1967), 352-362.
doi: 10.1007/BF00253152. |
[12] |
P. Germain, Q. S. Nguyen and P. Suquet,
Continuum thermodynamics, J. Appl. Mech., 50 (1983), 1010-1020.
|
[13] |
B. Halphen and Q. S. Nguyen,
Sur les matériaux standard généralisés, Journal de Mécanique, 14 (1975), 39-63.
|
[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. |
[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. |
[16] |
C. Licht,
Etude théorique et numérique de l'évolution d'un système fluide-flotteur, Thèse de docteur ingénieur, Nantes, 1980. |
[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. |
[18] |
C. Licht,
Evolution d'un système fluide-flotteur, Journal de Mécanique Théorique et Appliquée, 1 (1982), 211-235.
|
[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.
|
[20] |
C. Licht,
Comportement asymptotique d'une bande dissipative mince de faible rigidité, C. R. Acad. Sci. Paris, Ser. I, 317 (1993), 429-433.
|
[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. |
[22] |
C. Licht,
Thin linearly viscoelastic Kelvin-Voigt plates, C. R. Mecanique, 341 (2013), 697-700.
|
[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. |
[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. |
[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.
|
[26] |
J. L. Lions,
Réduction à des problèmes du type Cauchy-Kowaleska, Cours C.I.M.E., Cremonese, (1968), 269-280.
|
[27] |
J. L. Lions,
Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod, Gauthiers-Villars, Paris, 1968. |
[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. |
[29] |
H. F. Trotter,
Approximation of semi-groups of operators, Pacific J. Math., 8 (1958), 887-919.
doi: 10.2140/pjm.1958.8.887. |
[30] |
T. Weller,
Etude des symétries et modèles de plaques en piézoélectricité linéarisée, Thèse, Montpellier, 2004. |
[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. |
[32] |
T. Weller and C. Licht,
Asymptotic modeling of thin piezoelectric plates, Ann. Solid Struct. Mech., 1 (2010), 173-188.
|
[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. |
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