doi: 10.3934/cpaa.2021167
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Applied Analysis, 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

show all references

References:
[1]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

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) $
[1]

Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks & Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353

[2]

Markus Gahn, Maria Neuss-Radu, Peter Knabner. Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface. Networks & Heterogeneous Media, 2018, 13 (4) : 609-640. doi: 10.3934/nhm.2018028

[3]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631

[4]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485

[5]

Yangyang Shi, Hongjun Gao. Homogenization for stochastic reaction-diffusion equations with singular perturbation term. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021137

[6]

Yan-Yu Chen, Yoshihito Kohsaka, Hirokazu Ninomiya. Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 697-714. doi: 10.3934/dcdsb.2014.19.697

[7]

Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223

[8]

Simone Creo, Valerio Regis Durante. Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 65-90. doi: 10.3934/dcdss.2019005

[9]

Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151

[10]

Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic & Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65

[11]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016

[12]

Vitali Liskevich, Igor I. Skrypnik. Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1029-1042. doi: 10.3934/dcdss.2013.6.1029

[13]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405

[14]

Shin-Ichiro Ei, Hiroshi Matsuzawa. The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 901-921. doi: 10.3934/dcds.2010.26.901

[15]

Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems & Imaging, 2021, 15 (5) : 1015-1033. doi: 10.3934/ipi.2021026

[16]

Patrick De Kepper, István Szalai. An effective design method to produce stationary chemical reaction-diffusion patterns. Communications on Pure & Applied Analysis, 2012, 11 (1) : 189-207. doi: 10.3934/cpaa.2012.11.189

[17]

Danielle Hilhorst, Hideki Murakawa. Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium. Networks & Heterogeneous Media, 2014, 9 (4) : 669-682. doi: 10.3934/nhm.2014.9.669

[18]

Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks & Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143

[19]

Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523

[20]

Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (49)
  • HTML views (64)
  • Cited by (0)

Other articles
by authors

[Back to Top]