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Exploring Well-Posedness and Asymptotic Behavior in an Advection-Diffusion-Reaction (ADR) Model

Mohammed Elghandouri, Khalil Ezzinbi, Lamiae Saidi

TL;DR

This paper studies the Advection-Diffusion-Reaction ($ADR$) equation on bounded domains, establishing existence, uniqueness, and positivity of solutions and analyzing long-time behavior via a finite fractal dimensional global attractor using semigroup theory. It recasts the problem as an abstract evolution equation $u'(t)=\mathcal{A}u(t)+F(t,u(t))$ and proves global well-posedness under a structural hypothesis on reactions, then establishes the existence and finite fractal dimension of a global attractor. The authors derive an analytical 2-D solution by separation of variables and implement two explicit finite-difference schemes, a $2$-D centered difference and a $3$-D upwind method, to simulate ADR in 2D and 3D with stability, convergence, and error analyses plus attractor visualization. Numerical experiments model advection-dominated pollutant transport and NO–NO2–O3 chemistry, showing consistency with the theory and providing practical insights for environmental transport problems.

Abstract

In this paper, the existence, uniqueness, and positivity of solutions, as well as the asymptotic behavior through a finite fractal dimensional global attractor for a general Advection-Diffusion-Reaction (ADR) equation, are investigated. Our findings are innovative, as we employ semigroups and global attractors theories to achieve these results. Also, an analytical solution of a two-dimensional Advection-Diffusion Equation is presented. And finally, two Explicit Finite Difference schemes are used to simulate solutions in the two- and three-dimensional cases. The numerical simulations are conducted with predefined initial and Dirichlet boundary conditions.

Exploring Well-Posedness and Asymptotic Behavior in an Advection-Diffusion-Reaction (ADR) Model

TL;DR

This paper studies the Advection-Diffusion-Reaction () equation on bounded domains, establishing existence, uniqueness, and positivity of solutions and analyzing long-time behavior via a finite fractal dimensional global attractor using semigroup theory. It recasts the problem as an abstract evolution equation and proves global well-posedness under a structural hypothesis on reactions, then establishes the existence and finite fractal dimension of a global attractor. The authors derive an analytical 2-D solution by separation of variables and implement two explicit finite-difference schemes, a -D centered difference and a -D upwind method, to simulate ADR in 2D and 3D with stability, convergence, and error analyses plus attractor visualization. Numerical experiments model advection-dominated pollutant transport and NO–NO2–O3 chemistry, showing consistency with the theory and providing practical insights for environmental transport problems.

Abstract

In this paper, the existence, uniqueness, and positivity of solutions, as well as the asymptotic behavior through a finite fractal dimensional global attractor for a general Advection-Diffusion-Reaction (ADR) equation, are investigated. Our findings are innovative, as we employ semigroups and global attractors theories to achieve these results. Also, an analytical solution of a two-dimensional Advection-Diffusion Equation is presented. And finally, two Explicit Finite Difference schemes are used to simulate solutions in the two- and three-dimensional cases. The numerical simulations are conducted with predefined initial and Dirichlet boundary conditions.
Paper Structure (22 sections, 17 theorems, 92 equations, 7 figures)

This paper contains 22 sections, 17 theorems, 92 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Formulation of the model
  3. Origins and Problematic
  4. Description of the model
  5. Basic working tools
  6. Well-posedness
  7. Global attractor
  8. Why do we need to look for an attractor for equation \ref{['Eq 1']}?
  9. Existence of a global attractor for equation \ref{['Eq 1']}
  10. Finite fractal dimensional
  11. Analytical solution of an Advection-Diffusion Equation in 2-D using separation of variables
  12. Analytical solution in 2-D
  13. Analytical Example
  14. Numerical Simulations with Finite Difference Methods
  15. Numerical simulations in 2-D using explicit centered difference method
  16. ...and 7 more sections

Key Result

Theorem 3.2

Batkai A linear operator $A$ on a real Hilbert lattice space $H$ is dispersive if and only if $\langle Af,f^+\rangle_{H}\leq 0$ for every $f\in D(A)$.

Figures (7)

  • Figure 1: Analytical solutions at times 0, 0.05, 0.09, 0.12, with $u=5$ and $k=0.5$.
  • Figure 2: Numerical solutions at times 0, 0.05, 0.09, 0.12, with $u=5$ and $k=0.5$.
  • Figure 3: Maximum error between Analytical and Numerical solutions at times 0, 0.05, 0.09, 0.12, with $u=5$, $k=0.5$, and $m=n=40$.
  • Figure 4: Horizontal slice plots at $z = 1$ showing $\mathrm{NO}_2$ concentrations at times $0 \, \text{s}, 20 \, \text{s}, 50 \, \text{s}, 100 \, \text{s}, 200 \, \text{s}, 300 \, \text{s}, 400 \, \text{s}, 500 \, \text{s},$ and $600 \, \text{s}$, with $u = 1 \, \text{m/s}$ and $k = 0.00002 \, \text{m}^2/\text{s}$.
  • Figure 5: Horizontal slice plots at $z=1$ showing $\mathrm{O}_3$ concentrations at times $0 \, \text{s}, 20 \, \text{s}, 50 \, \text{s}, 100 \, \text{s}, 200 \, \text{s}, 300 \, \text{s}, 400 \, \text{s}, 500 \, \text{s},$ and $600 \, \text{s}$, with $u = 1 \, \text{m/s}$ and $k = 0.00002 \, \text{m}^2/\text{s}$.
  • ...and 2 more figures

Theorems & Definitions (32)

  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Theorem 3.4
  • Definition 3.5
  • Theorem 3.6
  • Corollary 3.7
  • Remark 3.8
  • Lemma 3.9
  • Lemma 3.10
  • ...and 22 more