January  2022, 21(1): 61-82. doi: 10.3934/cpaa.2021167

Singular limit for reactive transport through a thin heterogeneous layer including a nonlinear diffusion coefficient

Interdisciplinary Center for Scientific Computing, University of Heidelberg, Im Neuenheimer Feld 205, Heidelberg, 69120, Germany

Received  May 2021 Revised  August 2021 Published  January 2022 Early access  September 2021

Fund Project: The work of the author was supported by the Odysseus program of the Research Foundation - Flanders FWO (Project-Nr. G0G1316N) and the project SCIDATOS (Scientific Computing for Improved Detection and Therapy of Sepsis), which was funded by the Klaus Tschira Foundation, Germany (Grant number 00.0277.2015)

Reactive transport processes in porous media including thin heterogeneous layers play an important role in many applications. In this paper, we investigate a reaction-diffusion problem with nonlinear diffusion in a domain consisting of two bulk domains which are separated by a thin layer with a periodic heterogeneous structure. The thickness of the layer, as well as the periodicity within the layer are of order $ \epsilon $, where $ \epsilon $ is much smaller than the size of the bulk domains. For the singular limit $ \epsilon \to 0 $, when the thin layer reduces to an interface, we rigorously derive a macroscopic model with effective interface conditions between the two bulk domains. Due to the oscillations within the layer, we have the combine dimension reduction techniques with methods from the homogenization theory. To cope with these difficulties, we make use of the two-scale convergence in thin heterogeneous layers. However, in our case the diffusion in the thin layer is low and depends nonlinearly on the concentration itself. The low diffusion leads to a two-scale limit depending on a macroscopic and a microscopic variable. Hence, weak compactness results based on standard a priori estimates are not enough to pass to the limit $ \epsilon \to 0 $ in the nonlinear terms. Therefore, we derive strong two-scale compactness results based on a variational principle. Further, we establish uniqueness for the microscopic and the macroscopic model.

Citation: Markus Gahn. Singular limit for reactive transport through a thin heterogeneous layer including a nonlinear diffusion coefficient. Communications on Pure and Applied Analysis, 2022, 21 (1) : 61-82. doi: 10.3934/cpaa.2021167
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.

[2]

B. AmazianeL. Pankratov and V. Prytula, Homogenization of one phase flow in a highly heterogeneous porous medium including a thin layer, Asympt. Anal., 70 (2021), 51-86. 

[3]

N. S. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media, Kluwer Academic Publishers, 1989. doi: 10.1007/978-94-009-2247-1.

[4]

A. BourgeatO. Gipouloux and E. Marušić-Paloka, Mathematical modelling and numerical simulation of a non-{N}ewtonian viscous flow through a thin filter, SIAM J. Appl. Math., 62 (2001), 597-626.  doi: 10.1137/S0036139999354741.

[5]

A. BourgeatO. Gipouloux and E. Marušić-Paloka, Modelling of an underground waste disposal site by upscaling, Math. Meth. Appl. Sci., 27 (2004), 381-403.  doi: 10.1002/mma.459.

[6]

A. BourgeatS. Luckhaus and A. Mikelić, Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow, SIAM J. Math. Anal., 27 (1996), 1520-1543.  doi: 10.1137/S0036141094276457.

[7]

G. W. Clark and R. E. Showalter, Two-scale convergence of a model for flow in a partially fissured medium, Electron. J. Differ. Equ., 1999 (1999), 1-20. 

[8]

L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998.

[9]

M. GahnM. Neuss-Radu and P. Knabner, Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity, Discret. Contin. Dynam. Syst. Series S, 10 (2017), 773-797.  doi: 10.3934/dcdss.2017039.

[10]

M. GahnM. Neuss-Radu and P. Knabner, Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface, Networks and Heterogeneous Media, 13 (2018), 609-640.  doi: 10.3934/nhm.2018028.

[11]

A. Gaudiello and T. Mel'nyk, Homogenization of a nonlinear monotone problem with nonlinear signorini boundary conditions in a domain with highly rough boundary, J. Differ. Equ., 265 (2018), 5419-5454.  doi: 10.1016/j.jde.2018.07.002.

[12]

S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb, 10 (1970), 217-243. 

[13]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

[14]

F. List, K. Kumar, I. S. Pop and F. A. Radu, Upscaling of unsaturated flow in fractured porous media, arXiv, 2018. doi: 10.1137/18M1203754.

[15] J. MálekJ. NečasM. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, CRC Press, 1996. 
[16]

S. Marušić and E. Marušić-Paloka, Two-scale convergence for thin domains and its applications to some lower-dimensional model in fluid mechanics, Asympt. Anal., 23 (2000), 23-58. 

[17]

M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39 (2007), 687-720.  doi: 10.1137/060665452.

[18]

M. Neuss-RaduS. Ludwig and W. Jäger, Multiscale analysis and simulation of a reaction-diffusion problem with transmission conditions, Nonlinear Anal. Real World Appl., 11 (2010), 4572-4585.  doi: 10.1016/j.nonrwa.2008.11.024.

[19]

F. Otto, $L^1$-Contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differ. Equ., 131 (1996), 20-38.  doi: 10.1006/jdeq.1996.0155.

[20]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Springer-Verlag Berlin Heidelberg, 1980.

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]

B. AmazianeL. Pankratov and V. Prytula, Homogenization of one phase flow in a highly heterogeneous porous medium including a thin layer, Asympt. Anal., 70 (2021), 51-86. 

[3]

N. S. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media, Kluwer Academic Publishers, 1989. doi: 10.1007/978-94-009-2247-1.

[4]

A. BourgeatO. Gipouloux and E. Marušić-Paloka, Mathematical modelling and numerical simulation of a non-{N}ewtonian viscous flow through a thin filter, SIAM J. Appl. Math., 62 (2001), 597-626.  doi: 10.1137/S0036139999354741.

[5]

A. BourgeatO. Gipouloux and E. Marušić-Paloka, Modelling of an underground waste disposal site by upscaling, Math. Meth. Appl. Sci., 27 (2004), 381-403.  doi: 10.1002/mma.459.

[6]

A. BourgeatS. Luckhaus and A. Mikelić, Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow, SIAM J. Math. Anal., 27 (1996), 1520-1543.  doi: 10.1137/S0036141094276457.

[7]

G. W. Clark and R. E. Showalter, Two-scale convergence of a model for flow in a partially fissured medium, Electron. J. Differ. Equ., 1999 (1999), 1-20. 

[8]

L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998.

[9]

M. GahnM. Neuss-Radu and P. Knabner, Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity, Discret. Contin. Dynam. Syst. Series S, 10 (2017), 773-797.  doi: 10.3934/dcdss.2017039.

[10]

M. GahnM. Neuss-Radu and P. Knabner, Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface, Networks and Heterogeneous Media, 13 (2018), 609-640.  doi: 10.3934/nhm.2018028.

[11]

A. Gaudiello and T. Mel'nyk, Homogenization of a nonlinear monotone problem with nonlinear signorini boundary conditions in a domain with highly rough boundary, J. Differ. Equ., 265 (2018), 5419-5454.  doi: 10.1016/j.jde.2018.07.002.

[12]

S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb, 10 (1970), 217-243. 

[13]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

[14]

F. List, K. Kumar, I. S. Pop and F. A. Radu, Upscaling of unsaturated flow in fractured porous media, arXiv, 2018. doi: 10.1137/18M1203754.

[15] J. MálekJ. NečasM. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, CRC Press, 1996. 
[16]

S. Marušić and E. Marušić-Paloka, Two-scale convergence for thin domains and its applications to some lower-dimensional model in fluid mechanics, Asympt. Anal., 23 (2000), 23-58. 

[17]

M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39 (2007), 687-720.  doi: 10.1137/060665452.

[18]

M. Neuss-RaduS. Ludwig and W. Jäger, Multiscale analysis and simulation of a reaction-diffusion problem with transmission conditions, Nonlinear Anal. Real World Appl., 11 (2010), 4572-4585.  doi: 10.1016/j.nonrwa.2008.11.024.

[19]

F. Otto, $L^1$-Contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differ. Equ., 131 (1996), 20-38.  doi: 10.1006/jdeq.1996.0155.

[20]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Springer-Verlag Berlin Heidelberg, 1980.

Figure 1.  The microscopic domain $ \Omega_{\epsilon} $ containing the thin layer $ \Omega_{\epsilon} $ with periodic structure for $ n = 2 $. The heterogeneous structure for the thin layer is modeled by the oscillating diffusion coefficient $ D\left( \frac{x}{\epsilon}\right) $
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