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Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces
Existence and regularity of solutions for an evolution model of perfectly plastic plates
1. | Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy |
2. | Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy |
3. | Dipartimento di Matematica Guido Castelnuovo, Università degli Studi di Roma "La Sapienza", Piazzale Aldo Moro 5, 00185, Roma, Italy |
We continue the study of a dynamic evolution model for perfectly plastic plates, recently derived in [
References:
[1] |
H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984. |
[2] |
J. F. Babadjian and M. G. Mora,
Stress regularity in quasi-static perfect plasticity with a pressure dependent yield criterion, Journal of Differential Equations, 264 (2018), 5109-5151.
doi: 10.1016/j.jde.2017.12.034. |
[3] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. |
[4] |
A. Bensoussan and J. Frehse,
Asymptotic behaviour of the time-dependent Norton Hoff law in plasticity theory and $H^1$ regularity, Comment. Math. Univ. Carolinae, 37 (1996), 285-304.
|
[5] |
H. Brezis, Opérateurs maximaux monotones et semi-groupes de constractions dans les espaces de Hilbert, American Elsevier Publishing Co., Inc., New York, 1973. |
[6] |
P. Ciarlet, Mathematical Elasticity. Vol II. Theory of Plates, Studies in Mathematics and its Applications, 27. North-Holland Publishing Co., Amsterdam, 1997. |
[7] |
G. Dal Maso, A. DeSimone and M. G. Mora,
Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), 237-291.
doi: 10.1007/s00205-005-0407-0. |
[8] |
E. Davoli and M. G. Mora,
A quasistatic evolution model for perfectly plastic plates derived by $\Gamma$-convergence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 615-660.
doi: 10.1016/j.anihpc.2012.11.001. |
[9] |
E. Davoli and M. G. Mora,
Stress regularity for a new quasistatic evolution model of perfectly plastic plates, Calc. Var. Partial Differential Equations, 54 (2015), 2581-2614.
doi: 10.1007/s00526-015-0876-4. |
[10] |
F. Demengel, Fonctions à hessien borné, Ann. Inst. Fourier (Grénoble), 34 (1984), 155-190. |
[11] |
A. Demyanov,
Regularity of stresses in Prandtl-Reuss plasticity, Calc. Var. Partial Differential Equations, 34 (2009), 23-72.
doi: 10.1007/s00526-008-0174-5. |
[12] |
A. Demyanov,
Quasistatic evolution in the theory of perfectly elasto-plastic plates. Ⅰ. Existence of a weak solution, Math. Models Methods Appl. Sci., 19 (2009), 229-256.
doi: 10.1142/S0218202509003413. |
[13] |
A. Demyanov,
Quasistatic evolution in the theory of perfectly elasto-plastic plates. Ⅱ. Regularity of bending moments, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2137-2163.
doi: 10.1016/j.anihpc.2009.01.006. |
[14] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics Appl. Math., vol. 28, SIAM, Philadelphia, PA, 1999.
doi: 10.1137/1.9781611971088. |
[15] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer, 2011.
doi: 10.1007/978-0-387-09620-9. |
[16] |
R. V. Kohn and R. Temam,
Dual spaces of stresses and strains, with application to Hencky plasticity, Appl. Math. Optim., 10 (1983), 1-35.
doi: 10.1007/BF01448377. |
[17] |
J.Lubliner, Plasticity Theory, Macmillan Publishing Company, New York, 1990. Google Scholar |
[18] |
A. Mainik and A. Mielke,
Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.
doi: 10.1007/s00526-004-0267-8. |
[19] |
G. B. Maggiani and M. G. Mora,
A dynamic evolution model for perfectly plastic plates, Math. Models Methods Appl. Sci., 26 (2016), 1825-1864.
doi: 10.1142/S0218202516500469. |
[20] |
A. Mielke and T. Roubíček, Rate-independent Systems. Theory and Application, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2706-7. |
[21] |
R. T. Rockafellar, Convex Integral Functionals and Duality, in Contributions to Nonlinear Functional Analysis, Academic Press, (1971), 215-236. |
[22] |
P. M. Suquet, Sur le équations de la plasticité: existence et regularité des solutions, J. Mécanique, 20 (1981), 3-39. |
[23] |
R. Temam, Mathematical Problems in Plasticity, Gauthier-Villars, Paris, 1985. |
show all references
References:
[1] |
H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984. |
[2] |
J. F. Babadjian and M. G. Mora,
Stress regularity in quasi-static perfect plasticity with a pressure dependent yield criterion, Journal of Differential Equations, 264 (2018), 5109-5151.
doi: 10.1016/j.jde.2017.12.034. |
[3] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. |
[4] |
A. Bensoussan and J. Frehse,
Asymptotic behaviour of the time-dependent Norton Hoff law in plasticity theory and $H^1$ regularity, Comment. Math. Univ. Carolinae, 37 (1996), 285-304.
|
[5] |
H. Brezis, Opérateurs maximaux monotones et semi-groupes de constractions dans les espaces de Hilbert, American Elsevier Publishing Co., Inc., New York, 1973. |
[6] |
P. Ciarlet, Mathematical Elasticity. Vol II. Theory of Plates, Studies in Mathematics and its Applications, 27. North-Holland Publishing Co., Amsterdam, 1997. |
[7] |
G. Dal Maso, A. DeSimone and M. G. Mora,
Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), 237-291.
doi: 10.1007/s00205-005-0407-0. |
[8] |
E. Davoli and M. G. Mora,
A quasistatic evolution model for perfectly plastic plates derived by $\Gamma$-convergence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 615-660.
doi: 10.1016/j.anihpc.2012.11.001. |
[9] |
E. Davoli and M. G. Mora,
Stress regularity for a new quasistatic evolution model of perfectly plastic plates, Calc. Var. Partial Differential Equations, 54 (2015), 2581-2614.
doi: 10.1007/s00526-015-0876-4. |
[10] |
F. Demengel, Fonctions à hessien borné, Ann. Inst. Fourier (Grénoble), 34 (1984), 155-190. |
[11] |
A. Demyanov,
Regularity of stresses in Prandtl-Reuss plasticity, Calc. Var. Partial Differential Equations, 34 (2009), 23-72.
doi: 10.1007/s00526-008-0174-5. |
[12] |
A. Demyanov,
Quasistatic evolution in the theory of perfectly elasto-plastic plates. Ⅰ. Existence of a weak solution, Math. Models Methods Appl. Sci., 19 (2009), 229-256.
doi: 10.1142/S0218202509003413. |
[13] |
A. Demyanov,
Quasistatic evolution in the theory of perfectly elasto-plastic plates. Ⅱ. Regularity of bending moments, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2137-2163.
doi: 10.1016/j.anihpc.2009.01.006. |
[14] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics Appl. Math., vol. 28, SIAM, Philadelphia, PA, 1999.
doi: 10.1137/1.9781611971088. |
[15] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer, 2011.
doi: 10.1007/978-0-387-09620-9. |
[16] |
R. V. Kohn and R. Temam,
Dual spaces of stresses and strains, with application to Hencky plasticity, Appl. Math. Optim., 10 (1983), 1-35.
doi: 10.1007/BF01448377. |
[17] |
J.Lubliner, Plasticity Theory, Macmillan Publishing Company, New York, 1990. Google Scholar |
[18] |
A. Mainik and A. Mielke,
Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.
doi: 10.1007/s00526-004-0267-8. |
[19] |
G. B. Maggiani and M. G. Mora,
A dynamic evolution model for perfectly plastic plates, Math. Models Methods Appl. Sci., 26 (2016), 1825-1864.
doi: 10.1142/S0218202516500469. |
[20] |
A. Mielke and T. Roubíček, Rate-independent Systems. Theory and Application, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2706-7. |
[21] |
R. T. Rockafellar, Convex Integral Functionals and Duality, in Contributions to Nonlinear Functional Analysis, Academic Press, (1971), 215-236. |
[22] |
P. M. Suquet, Sur le équations de la plasticité: existence et regularité des solutions, J. Mécanique, 20 (1981), 3-39. |
[23] |
R. Temam, Mathematical Problems in Plasticity, Gauthier-Villars, Paris, 1985. |
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