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Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line
Homogenization and singular perturbation in porous media
Departmant of mathematics, University of Zagreb, Bijenička 30, 10000, Zagreb, Croatia |
We study a Dirichlet problem in periodic porous medium depending on two small parameters, the hydraulic permeability of the porous inclusions $ \delta $ and the period $ \varepsilon $. We study the situation as $ \delta\to 0 \;, \; \varepsilon\to 0 $ and $ \varepsilon \to 0 \;, \; \delta\to 0 $ and prove that the two limits do not commute.
References:
| [1] |
G. Allaire,
Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
| [2] |
G. Allaire,
Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. Ⅰ. Abstract framework, a volume distribution of holes, Arch. Ration. Mech. Anal., 113 (1990), 209-259.
doi: 10.1007/BF00375065. |
| [3] |
T. Arbogast, J. Jr. Douglas and U. Hornung,
Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836.
doi: 10.1137/0521046. |
| [4] |
J. M. Arrieta and M. Pereira,
Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures Appl., 96 (2011), 29-57.
doi: 10.1016/j.matpur.2011.02.003. |
| [5] |
N. S. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Porous Media Periodic, Springer, 1989.
doi: 10.1007/978-94-009-2247-1. |
| [6] |
F. A. Bornemann, Homogenization in Time of Singularly Perturbed Conservative Mechanical Systems, Springer, 1998.
doi: 10.1007/BFb0092091. |
| [7] |
A. Bourgeat, E. Marušić-Paloka and A. Mikelić,
Effective Fluid Flow in a Porous Medium Containing a Thin Fissure, Asymptotic Analysis, 11 (1995), 241-262.
|
| [8] |
A. Bourgeat, S. Luckhaus and A. Mikelić,
A rigorous result for a double porosity model of immiscible two-phase flows, SIAM J. Math. Anal., 27 (1996), 1520-1543.
doi: 10.1137/S0036141094276457. |
| [9] |
D. Caillerie,
Homogénéisation des équations de la diffusion stationnaire dans les domaines cylindriques aplatis, RAIRO Analyse Numeérique, 15 (1981), 295-319.
|
| [10] |
D. Caillerie, Thin elastic and perodic plates, Math. Methods Appl. Sci., 6 (1984), 159-191.
doi: 10.1002/mma.1670060112. |
| [11] |
G. S. Chechkin,
Averaging of boundary value problems with a singular perturbation of the boundary conditions, Mat. Sb., 186 (1993), 191-222.
doi: 10.1070/SM1994v079n01ABEH003608. |
| [12] |
D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer, 1999.
doi: 10.1007/978-1-4612-2158-6. |
| [13] |
D. Cioranescu, P. Donato and R. Zaki,
The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2016), 467-496.
|
| [14] |
A. Damlamian and M. Vogelius,
Homogenization limits of diffusion equations in thin domains, Mod. Math. An. Num. M2AN, 22 (1988), 53-74.
doi: 10.1051/m2an/1988220100531. |
| [15] |
A. Damlamian and M. Vogelius,
Homogenization limits of the eequations of elasticity in thin domains, SIAM J. Math. Anal., 18 (1987), 435-451.
doi: 10.1137/0518034. |
| [16] |
T. Fratrović and E. Marušić-Paloka,
Nonlinear Brinkman-type law as a critical case in the polymer fluid filtration, Appl. Anal., 95 (2016), 562-583.
doi: 10.1080/00036811.2015.1022155. |
| [17] |
M. Jurak and Z. Tutek,
A one-dimensional model of homogenized rod, Glas. Mat., 24 (1989), 271-290.
|
| [18] |
R. Kohn and M. Vogelius,
A new model for thin plates with rapidly varying thickness, Ⅱ a convergence proof, Quart. Appl. Math., 43 (1985), 1-22.
doi: 10.1090/qam/782253. |
| [19] |
S. Marušić,
A note on permeability for a network of thin channels, Glas. Mat., 39 (2004), 339-346.
doi: 10.3336/gm.39.2.16. |
| [20] |
E. Marušić-Paloka and S. Marušić,
Computation of the permeability tensor for the fluid flow through a periodic net of thin channels, Appl. Anal., 64 (1997), 27-37.
doi: 10.1080/00036819708840521. |
| [21] |
E. Marušić-Paloka, I. Pažanin and S. Marušić,
Comparison between Darcy and Brinkman laws in a fracture, Appl. Math. Comput., 218 (2012), 7538-7545.
doi: 10.1016/j.amc.2012.01.021. |
| [22] |
G. A. Nguetseng,
A General convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.
doi: 10.1137/0520043. |
| [23] |
E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory, Springer, 1980. |
| [24] |
H. Kim and H. Shahgholian,
Homogenization of a singular perturbation problem, J. Math. Sci., 243 (2019), 163-176.
doi: 10.1007/s10958-019-04472-x. |
| [25] |
P. Shi, A. Spagnuolo and S. Wright,
Reiterated Homogenization and the Double-Porosity Model, Transport Porous Med., 59 (2005), 73-95.
doi: 10.1007/s11242-004-1121-3. |
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] |
G. Allaire,
Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. Ⅰ. Abstract framework, a volume distribution of holes, Arch. Ration. Mech. Anal., 113 (1990), 209-259.
doi: 10.1007/BF00375065. |
| [3] |
T. Arbogast, J. Jr. Douglas and U. Hornung,
Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836.
doi: 10.1137/0521046. |
| [4] |
J. M. Arrieta and M. Pereira,
Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures Appl., 96 (2011), 29-57.
doi: 10.1016/j.matpur.2011.02.003. |
| [5] |
N. S. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Porous Media Periodic, Springer, 1989.
doi: 10.1007/978-94-009-2247-1. |
| [6] |
F. A. Bornemann, Homogenization in Time of Singularly Perturbed Conservative Mechanical Systems, Springer, 1998.
doi: 10.1007/BFb0092091. |
| [7] |
A. Bourgeat, E. Marušić-Paloka and A. Mikelić,
Effective Fluid Flow in a Porous Medium Containing a Thin Fissure, Asymptotic Analysis, 11 (1995), 241-262.
|
| [8] |
A. Bourgeat, S. Luckhaus and A. Mikelić,
A rigorous result for a double porosity model of immiscible two-phase flows, SIAM J. Math. Anal., 27 (1996), 1520-1543.
doi: 10.1137/S0036141094276457. |
| [9] |
D. Caillerie,
Homogénéisation des équations de la diffusion stationnaire dans les domaines cylindriques aplatis, RAIRO Analyse Numeérique, 15 (1981), 295-319.
|
| [10] |
D. Caillerie, Thin elastic and perodic plates, Math. Methods Appl. Sci., 6 (1984), 159-191.
doi: 10.1002/mma.1670060112. |
| [11] |
G. S. Chechkin,
Averaging of boundary value problems with a singular perturbation of the boundary conditions, Mat. Sb., 186 (1993), 191-222.
doi: 10.1070/SM1994v079n01ABEH003608. |
| [12] |
D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer, 1999.
doi: 10.1007/978-1-4612-2158-6. |
| [13] |
D. Cioranescu, P. Donato and R. Zaki,
The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2016), 467-496.
|
| [14] |
A. Damlamian and M. Vogelius,
Homogenization limits of diffusion equations in thin domains, Mod. Math. An. Num. M2AN, 22 (1988), 53-74.
doi: 10.1051/m2an/1988220100531. |
| [15] |
A. Damlamian and M. Vogelius,
Homogenization limits of the eequations of elasticity in thin domains, SIAM J. Math. Anal., 18 (1987), 435-451.
doi: 10.1137/0518034. |
| [16] |
T. Fratrović and E. Marušić-Paloka,
Nonlinear Brinkman-type law as a critical case in the polymer fluid filtration, Appl. Anal., 95 (2016), 562-583.
doi: 10.1080/00036811.2015.1022155. |
| [17] |
M. Jurak and Z. Tutek,
A one-dimensional model of homogenized rod, Glas. Mat., 24 (1989), 271-290.
|
| [18] |
R. Kohn and M. Vogelius,
A new model for thin plates with rapidly varying thickness, Ⅱ a convergence proof, Quart. Appl. Math., 43 (1985), 1-22.
doi: 10.1090/qam/782253. |
| [19] |
S. Marušić,
A note on permeability for a network of thin channels, Glas. Mat., 39 (2004), 339-346.
doi: 10.3336/gm.39.2.16. |
| [20] |
E. Marušić-Paloka and S. Marušić,
Computation of the permeability tensor for the fluid flow through a periodic net of thin channels, Appl. Anal., 64 (1997), 27-37.
doi: 10.1080/00036819708840521. |
| [21] |
E. Marušić-Paloka, I. Pažanin and S. Marušić,
Comparison between Darcy and Brinkman laws in a fracture, Appl. Math. Comput., 218 (2012), 7538-7545.
doi: 10.1016/j.amc.2012.01.021. |
| [22] |
G. A. Nguetseng,
A General convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.
doi: 10.1137/0520043. |
| [23] |
E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory, Springer, 1980. |
| [24] |
H. Kim and H. Shahgholian,
Homogenization of a singular perturbation problem, J. Math. Sci., 243 (2019), 163-176.
doi: 10.1007/s10958-019-04472-x. |
| [25] |
P. Shi, A. Spagnuolo and S. Wright,
Reiterated Homogenization and the Double-Porosity Model, Transport Porous Med., 59 (2005), 73-95.
doi: 10.1007/s11242-004-1121-3. |
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