Well-posedness of a nonlinear stochastic model for a chemical reaction in porous media and applications

Mhamed Eddahbi; Mogtaba Mohammed; Hammou El-Otmany; Mhamed Eddahbi; Mogtaba Mohammed; Hammou El-Otmany

doi:10.3934/math.20241211

AIMS Mathematics

2024, Volume 9, Issue 9: 24860-24886. doi: 10.3934/math.20241211

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Well-posedness of a nonlinear stochastic model for a chemical reaction in porous media and applications

1.
Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Z.C. 11451, Riyadh, Saudi Arabia
2.
Department of Mathematics, College of Science, Majmaah University, Al-Majmaah 11952, Saudi Arabia
3.
Toulouse School of Economics, University of Toulouse Capitole 1, Esplanade de l'Université, 31080 Toulouse Cedex 6, France

Received: 11 June 2024 Revised: 01 August 2024 Accepted: 07 August 2024 Published: 26 August 2024
MSC : 60H15, 60H30, 60H35, 76V05

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.
- chemical reaction flows,
- nonlinear stochastic PDEs,
- Galerkin's approximation,
- positivity of solutions,
- numerical simulations
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

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Abstract

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.

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