Global Existence and Finite-Time Blow-Up for a Coupled Darcy-Forchheimer-Brinkman System with Quadratic Reaction Dynamics

TL;DR

The paper analyzes a fully coupled Darcy–Forchheimer–Brinkman system with a nonlinear convection–diffusion–reaction equation modeling reactive transport in porous media. It develops a Faedo–Galerkin framework with viscosity truncation and a maximum principle to prove local and global weak solutions in $d=2,3$ under $0 \le c_0 \le 1$, and establishes strong solutions for higher-regular initial data, along with exponential decay of the concentration in $L^p$ for all $1 \le p \le \infty$ and finite-time blow-up when $c_0>1$ (with explicit blow-up time bounds). A dichotomy in long-time behavior is derived: sub-threshold data yield global stability via $c\to 0$ exponentially, while super-threshold data lead to singularity in finite time. Numerical simulations using COMSOL corroborate the theoretical predictions, demonstrating both decay and blow-up, and underscoring the results’ relevance for porous-media reactive transport.

Abstract

We study a nonlinear system coupling the Darcy-Forchheimer-Brinkman equations with a convection-diffusion-reaction equation, arising in reactive transport through porous media. The model features a nonlinear viscosity coupling, Forchheimer inertial drag, convective transport, and a quadratic reaction term. We establish the existence of local-in-time weak solutions for general initial data. Under the physically relevant condition on initial data $0 \leq c_0 \leq 1$, a maximum principle for the concentration is proved, yielding global existence and uniqueness of weak solutions in two and three space dimensions. For higher regular initial data, we obtain the existence, uniqueness, and continuous dependence of strong solutions. In this regime, the concentration decays exponentially to zero in $L^p$-norm for all $1 \leq p \leq \infty$ with a uniform decay rate. In contrast, if $c_0 > 1$, we demonstrate the occurrence of finite-time blow-up of solutions and derive an explicit upper bound for the blow-up time. Finally, numerical simulations based on the finite element method are presented to illustrate both the decay behavior and finite-time blow-up predicted by the theory.

Figures (7)

• Figure 1: Time evolution of the $L^1(\Omega)$-norm of the concentration, $\|c(t)\|_{L^1(\Omega)}$, for different values of the reaction parameter $\kappa$, with $M_0 = 0.8$.
• Figure 2: Time evolution of the $L^1(\Omega)$-norm of the concentration, $\|c(t)\|_{L^\infty(\Omega)}$, for different values of the reaction parameter $\kappa$, with $M_0 = 0.8$.
• Figure 3: Numerical decay rate $\lambda_{\mathrm{num}}$ of the $L^1(\Omega)$-norm of the concentration, $\|c(t)\|_{L^1(\Omega)}$, for fixed $M_0 = 0.8$ and $\kappa = 0.005,\,0.01,\,0.02$.
• Figure 4: Numerical decay rate of the $L^1(\Omega)$-norm of the concentration, $\|c(t)\|_{L^1(\Omega)}$, for fixed $M_0 = 0.8$.
• Figure 5: Numerical decay rate $\lambda_{\mathrm{num}}$ of the $L^1(\Omega)$-norm of the concentration, $\|c(t)\|_{L^1(\Omega)}$, for fixed $\kappa = 0.01$ and $M_0 = 0.4,\,0.6,\,0.8$.

Theorems & Definitions (31)

• Theorem 3.1: Gagliardo–Nirenberg, cf. migorski2019nonmonotone, garcke2019
• Theorem 3.2: Aubin-Lions, cf. migorski2019nonmonotone
• Theorem 3.3: Young's inequality with $\epsilon$
• Theorem 3.4: Comparison principle for ODEs, cf. lakshmikantham1969differential
• Definition 3.1: Weak and Strong Solutions of System \ref{['model']}
• Theorem 4.1
• Lemma 4.1
• proof
• Lemma 4.2
• proof