Homogenization of diffusion equation with scalar hysteresis operator

Jan Franců

Mathematica Bohemica (2001)

  • Volume: 126, Issue: 2, page 363-377
  • ISSN: 0862-7959

Abstract

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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 ε and η ε when the spatial period ε tends to zero. The homogenized characteristics c * and η * are identified and the convergence of the corresponding solutions to the solution of the homogenized equation is proved.

How to cite

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Franců, Jan. "Homogenization of diffusion equation with scalar hysteresis operator." Mathematica Bohemica 126.2 (2001): 363-377. <http://eudml.org/doc/248882>.

@article{Franců2001,
abstract = {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.},
author = {Franců, Jan},
journal = {Mathematica Bohemica},
keywords = {hysteresis; Prandtl-Ishlinskii operator; material with periodic structure; nonlinear diffusion equation; homogenization; initial-boundary value problem; Prandtl-Ishlinskij operator; material with periodic structure; nonlinear diffusion equation; spatially periodic data},
language = {eng},
number = {2},
pages = {363-377},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Homogenization of diffusion equation with scalar hysteresis operator},
url = {http://eudml.org/doc/248882},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Franců, Jan
TI - Homogenization of diffusion equation with scalar hysteresis operator
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 2
SP - 363
EP - 377
AB - 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.
LA - eng
KW - hysteresis; Prandtl-Ishlinskii operator; material with periodic structure; nonlinear diffusion equation; homogenization; initial-boundary value problem; Prandtl-Ishlinskij operator; material with periodic structure; nonlinear diffusion equation; spatially periodic data
UR - http://eudml.org/doc/248882
ER -

References

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  8. Systems with Hysteresis, (Rusian edition: Nauka, Moskva, 1983). Springer, Berlin, 1989. (1989) MR0987431
  9. Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Vol. 8, Gakuto Int. Series Math. Sci. & Appl., Gakkötosho, Tokyo, 1996. (1996) MR2466538
  10. Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. Angew. Math. Mech. 8 (1928), 85–106. (German) (1928) 
  11. Differential Models of Hysteresis, Springer, Berlin, 1994. (1994) Zbl0820.35004MR1329094

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