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September  2019, 39(9): 5403-5429. doi: 10.3934/dcds.2019221

## Stochastic homogenization for a diffusion-reaction model

 1 Department of Mathematics & Statistics, University of Wyoming, East University Avenue, Laramie WY 82071, USA 2 Department of Mathematics & ISC, Texas A&M University, College Station, TX, USA 3 Multiscale Modeling Laboratory, North-Eastern Federal University, Yakutsk, 677980, Russia 4 Alexandru Ioan Cuza University of Iasi, Faculty of Economics and Business Administration, Bd. Carol I, 22, Iasi 700505, Romania

* Corresponding author

Received  November 2018 Published  May 2019

In this paper, we study stochastic homogenization of a coupled diffusion-reaction system. The diffusion-reaction system is coupled to stochastic differential equations, which govern the changes in the media properties. Though homogenization with changing media properties has been studied in previous findings, there is little research on homogenization when the media properties change due to stochastic differential equations. Such processes occur in many applications, where the changes in media properties are due to particle deposition. In the paper, we investigate the well-posedness of the nonlinear fine-grid (resolved) problem and derive limiting equations. We formulate the cell problems and derive the limiting equations, which are deterministic with nonlinear reaction terms. The limiting equations involve the invariant measures corresponding to stochastic differential equations. The obtained results can play an important role for modeling in porous media and allow the use of simplified and deterministic limiting equations.

Citation: Hakima Bessaih, Yalchin Efendiev, Razvan Florian Maris. Stochastic homogenization for a diffusion-reaction model. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5403-5429. doi: 10.3934/dcds.2019221
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084. [2] H. Bessaih, Y. Efendiev and F. Maris, Homogenization of Brinkman flows in heterogenous dynamic media, SPDE: Analysis and Computations, 3 (2015), 479-505.  doi: 10.1007/s40072-015-0058-6. [3] S. Cerrai, Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach, Lecture Notes in Mathematics, 1762. Springer-Verlag, Berlin, 2001. doi: 10.1007/b80743. [4] S. Cerrai, A Khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab., 19 (2009), 899-948.  doi: 10.1214/08-AAP560. [5] S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z. [6] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829. [7] M. Freidlin and A. Wentzell, Averaging principle for stochastic perturbations of multifrequency systems, Stochastics and Dynamics, 3 (2003), 393-408.  doi: 10.1142/S0219493703000747. [8] J. Jacod and P. Protter, Probability Essentials, Universitext, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-51431-9. [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [10] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and its Applications 2, North-Holland Publishing Co., Amsterdam-New York, 1979. [11] K. Yosida, Functional Analysis. Reprint of the sixth (1980) edition. Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8.

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] H. Bessaih, Y. Efendiev and F. Maris, Homogenization of Brinkman flows in heterogenous dynamic media, SPDE: Analysis and Computations, 3 (2015), 479-505.  doi: 10.1007/s40072-015-0058-6. [3] S. Cerrai, Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach, Lecture Notes in Mathematics, 1762. Springer-Verlag, Berlin, 2001. doi: 10.1007/b80743. [4] S. Cerrai, A Khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab., 19 (2009), 899-948.  doi: 10.1214/08-AAP560. [5] S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z. [6] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829. [7] M. Freidlin and A. Wentzell, Averaging principle for stochastic perturbations of multifrequency systems, Stochastics and Dynamics, 3 (2003), 393-408.  doi: 10.1142/S0219493703000747. [8] J. Jacod and P. Protter, Probability Essentials, Universitext, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-51431-9. [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [10] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and its Applications 2, North-Holland Publishing Co., Amsterdam-New York, 1979. [11] K. Yosida, Functional Analysis. Reprint of the sixth (1980) edition. Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8.
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