In this paper, we considered a stochastic model of chemical reactive flows acting through porous media under the influence of nonlinear external random fluctuations, where the interchanges of chemical flow across the skeleton's surface are represented by a nonlinear function. We studied the existence and uniqueness of strong probabilistic solutions for the model under consideration. We also show the positivity for the concentration of the solute in the fluid face as well as the concentration of reactants on the surface of the skeleton under extra reasonable assumptions on the data. Initially, we approximated the solution of the nonlinear stochastic diffusion equation using Galerkin's approximation, and obtained important bound estimates along with probabilistic compactness results. Thereafter, we passed the limit and obtained a weak probabilistic solution. This was followed by the path-wise uniqueness of the solution, which leads to the existence and uniqueness of strong probabilistic solutions as a result of Yamada-Watanabe's theorem. Finally, we discuss some important numerical applications such as Langmuir and Freundlich kinetics using the extended stochastic non-conforming finite element method to illustrate the efficiency of this approach and compare it to the deterministic approach in both cases. Let us mention that well-posedness, positivity, and numerical simulations have not been considered so far for such a nonlinear stochastic model.
Citation: Mhamed Eddahbi, Mogtaba Mohammed, Hammou El-Otmany. Well-posedness of a nonlinear stochastic model for a chemical reaction in porous media and applications[J]. AIMS Mathematics, 2024, 9(9): 24860-24886. doi: 10.3934/math.20241211
In this paper, we considered a stochastic model of chemical reactive flows acting through porous media under the influence of nonlinear external random fluctuations, where the interchanges of chemical flow across the skeleton's surface are represented by a nonlinear function. We studied the existence and uniqueness of strong probabilistic solutions for the model under consideration. We also show the positivity for the concentration of the solute in the fluid face as well as the concentration of reactants on the surface of the skeleton under extra reasonable assumptions on the data. Initially, we approximated the solution of the nonlinear stochastic diffusion equation using Galerkin's approximation, and obtained important bound estimates along with probabilistic compactness results. Thereafter, we passed the limit and obtained a weak probabilistic solution. This was followed by the path-wise uniqueness of the solution, which leads to the existence and uniqueness of strong probabilistic solutions as a result of Yamada-Watanabe's theorem. Finally, we discuss some important numerical applications such as Langmuir and Freundlich kinetics using the extended stochastic non-conforming finite element method to illustrate the efficiency of this approach and compare it to the deterministic approach in both cases. Let us mention that well-posedness, positivity, and numerical simulations have not been considered so far for such a nonlinear stochastic model.
[1] | Z. I. Ali, M. Sango, Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators, Ukr. Math. J., 74 (2022), 997–1020. https://doi.org/10.1007/s11253-022-02117-y doi: 10.1007/s11253-022-02117-y |
[2] | G. Allaire, H. Hutridurga, Homogenization of reactive flows in porous media and competition between bulk and surface diffusion, IMA J. Appl. Math., 77 (2012), 788–815. https://doi.org/10.1093/imamat/hxs049 doi: 10.1093/imamat/hxs049 |
[3] | A. Bensoussan, Some existence results for stochastic partial differential equations, Pitman Res. Notes Math. Ser., 268 (1992), 37–53. |
[4] | H. Bessaih, Y. Efendiev, R. F. Maris, Stochastic homogenization of a convection-diffusion equation, SIAM J. Math. Anal., 53 (2021), 2718–2745. https://doi.org/10.1137/19M130277 doi: 10.1137/19M130277 |
[5] | P. Billingsley, Convergence of probability measures, John Wiley & Sons, 2013. |
[6] | B. Cabarrubias, P. Donato, Existence and uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions, Carpathian J. Math., 27 (2011), 173–184. |
[7] | I. Chourabi, P. Donato, Bounded solutions for a quasilinear singular problem with nonlinear Robin boundary conditions, Differ. Integral Equ., 26 (2013), 975–1008. https://doi.org/10.57262/die/1372858558 doi: 10.57262/die/1372858558 |
[8] | I. Chourabi, P. Donato, Homogenization of elliptic problems with quadratic growth and nonhomogenous Robin conditions in perforated domains, Chin. Ann. Math. Ser. B, 37 (2016), 833–852. https://doi.org/10.1007/s11401-016-1008-y doi: 10.1007/s11401-016-1008-y |
[9] | P. L. Chow, Stochastic partial differential equations, CRC Press, 2014. |
[10] | C. Conca, J. I. Díaz, A. Linan, C. Timofte, Homogenization in chemical reactive flows, Electron. J. Differ. Equ., 2004 (2004), 1–22. |
[11] | C. Conca, J. I. Díaz, C. Timofte, Effective chemical processes in porous media, Math. Models Methods Appl. Sci., 13 (2003), 1437–1462. https://doi.org/10.1142/S0218202503002982 doi: 10.1142/S0218202503002982 |
[12] | L. Denis, A. Matoussi, L. Stoica, Maximum principle and comparison theorem for quasi-linear stochastic PDE's, Electron. Commun. Probab., 14 (2009), 500–530. https://doi.org/10.1214/EJP.v14-629 doi: 10.1214/EJP.v14-629 |
[13] | P. Donato, S. Monsurrò, F. Raimondi, Existence and uniqueness results for a class of singular elliptic problems in perforated domains, Ric. Mat., 66 (2017), 333–360. https://doi.org/10.1007/s11587-016-0303-y doi: 10.1007/s11587-016-0303-y |
[14] | G. Ferreyra, P. Sundar, Comparison of solutions of stochastic equations and applications, Stoch. Anal. Appl., 18 (2000), 211–229. https://doi.org/10.1080/07362990008809665 doi: 10.1080/07362990008809665 |
[15] | U. Hornung, W. Jäer, Diffusion, convection, adsorption, and reaction of chemicals in porous media, J. Differ. Equ., 92 (1991), 199–225. https://doi.org/10.1016/0022-0396(91)90047-D doi: 10.1016/0022-0396(91)90047-D |
[16] | M. Mohammed, Well-posedness for nonlinear parabolic stochastic differential equations with nonlinear Robin conditions, Symmetry, 14 (2022), 1–19. https://doi.org/10.3390/sym14081722 doi: 10.3390/sym14081722 |
[17] | M. Mohammed, Homogenization of a nonlinear stochastic model with nonlinear random forces for chemical reactive flows in porous media, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 4598–4624. https://doi.org/10.3934/dcdsb.2023032 doi: 10.3934/dcdsb.2023032 |
[18] | M. Mohammed, N. Ahmed, Homogenization and correctors of Robin problem for linear stochastic equations in periodically perforated domains, Asymptot. Anal., 120 (2020), 123–149. https://doi.org/10.3233/ASY-191582 doi: 10.3233/ASY-191582 |
[19] | M. Mohammed, M. Sango, Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: the two scale convergence method, Asymptot. Anal., 91 (2015), 341–371. https://doi.org/10.3233/ASY-141269 doi: 10.3233/ASY-141269 |
[20] | M. Mohammed, M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptot. Anal., 97 (2016), 301–327. https://doi.org/10.3233/ASY-151355 doi: 10.3233/ASY-151355 |
[21] | M. Mohammed, M. Sango, A Tartar approach to periodic homogenization of linear hyperbolic stochastic partial differential equation, Int. J. Modern Phys. B, 30 (2016), 1640020. https://doi.org/10.1142/S0217979216400208 doi: 10.1142/S0217979216400208 |
[22] | M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math., 426 (2004), 1–63. |
[23] | B. L. Rozovsky, S. V. Lototsky, Stochastic evolution systems: linear theory and applications to non-linear filtering, Cham: Springer, 2018. https://doi.org/10.1007/978-3-319-94893-5 |
[24] | M. Sango, Splitting-up scheme for nonlinear stochastic hyperbolic equations, Forum Math., 25 (2013), 931–965. https://doi.org/10.1515/form.2011.138 doi: 10.1515/form.2011.138 |
[25] | M. Sango, Stochastic Navier-Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability, Ukr. Math. J., 75 (2023), 600–620. https://doi.org/10.1007/s11253-023-02219-1 doi: 10.1007/s11253-023-02219-1 |