| Title:
|
Homogenization of diffusion equation with scalar hysteresis operator (English) |
| Author:
|
Franců, Jan |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
126 |
| Issue:
|
2 |
| Year:
|
2001 |
| Pages:
|
363-377 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper deals with a scalar diffusion equation $ c\,u_t = ({{F}}[u_x])_x + f, $ where ${F}$ is a Prandtl-Ishlinskii operator and $c, f$ are given functions. In the diffusion or heat conduction equation the linear constitutive relation is replaced by a scalar Prandtl-Ishlinskii hysteresis spatially dependent operator. We prove existence, uniqueness and regularity of solution to the corresponding initial-boundary value problem. The problem is then homogenized by considering a sequence of equations of the above type with spatially periodic data $c^\varepsilon $ and $\eta ^\varepsilon $ when the spatial period $\varepsilon $ tends to zero. The homogenized characteristics $c^*$ and $\eta ^*$ are identified and the convergence of the corresponding solutions to the solution of the homogenized equation is proved. (English) |
| Keyword:
|
hysteresis |
| Keyword:
|
Prandtl-Ishlinskii operator |
| Keyword:
|
material with periodic structure |
| Keyword:
|
nonlinear diffusion equation |
| Keyword:
|
homogenization |
| Keyword:
|
initial-boundary value problem |
| Keyword:
|
spatially periodic data |
| MSC:
|
34C55 |
| MSC:
|
35B27 |
| MSC:
|
47J40 |
| idZBL:
|
Zbl 0977.35017 |
| idMR:
|
MR1844275 |
| DOI:
|
10.21136/MB.2001.134031 |
| . |
| Date available:
|
2009-09-24T21:51:33Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134031 |
| . |
| Reference:
|
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| Reference:
|
[2] A. Bensoussan, J. L. Lions, G. Papanicolaou: Asymptotic Analysis for Periodic Structure.North Holland, 1978. MR 0503330 |
| Reference:
|
[3] M. Brokate: Hysteresis operators.Phase Transitions and Hysteresis, Lecture Notes of C.I.M.E., Montecatini Terme, July 13–21, 1993, Springer, Berlin, 1994, pp. 1–38. Zbl 0836.35065, MR 1321830 |
| Reference:
|
[4] M. Brokate, J. Sprekels: Hysteresis and Phase Transitions.Appl. Math. Sci., Vol. 121, Springer, New York, 1996. MR 1411908 |
| Reference:
|
[5] J. Franců, P. Krejčí: Homogenization of scalar wave equation with hysteresis.Continuum Mech. Thermodyn. 11 (1999), 371–390. MR 1727713, 10.1007/s001610050118 |
| Reference:
|
[6] J. Franců: Homogenization of scalar hysteresis operators.Equadiff 9 CD ROM, Masaryk University, Brno, 1998, pp. 111–122. |
| Reference:
|
[7] A. Yu. Ishlinskii: Some applications of statistical methods to describing deformations of bodies.Izv. AN SSSR, Techn. Ser. 9 (1944), 580–590. (Russian) |
| Reference:
|
[8] M. A. Krasnosel’skii, A.V. Pokrovskii: Systems with Hysteresis.(Rusian edition: Nauka, Moskva, 1983). Springer, Berlin, 1989. MR 0987431 |
| Reference:
|
[9] P. Krejčí: Hysteresis, Convexity and Dissipation in Hyperbolic Equations.Vol. 8, Gakuto Int. Series Math. Sci. & Appl., Gakkötosho, Tokyo, 1996. MR 2466538 |
| Reference:
|
[10] L. Prandtl: Ein Gedankenmodell zur kinetischen Theorie der festen Körper.Z. Angew. Math. Mech. 8 (1928), 85–106. (German) 10.1002/zamm.19280080202 |
| Reference:
|
[11] A. Visintin: Differential Models of Hysteresis.Springer, Berlin, 1994. Zbl 0820.35004, MR 1329094 |
| . |
