Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reaction-Diffusion System Approximation to the Fast Diffusion Equation

Hideki Murakawa, Florian Salin

Abstract

This paper proposes a novel reaction-diffusion system approximation tailored for singular diffusion problems, typified by the fast diffusion equation. While such approximation methods have been successfully applied to degenerate parabolic equations, their extension to singular diffusion-where the diffusion coefficient diverges at low densities-has remained unexplored. To address this, we construct an approximating semilinear system characterized by a reaction relaxation parameter and a time-derivative regularizing parameter. We rigorously establish the well-posedness of this system and derive uniform a priori estimates. Using compactness arguments, we prove the convergence of the approximate solutions to the unique weak solution of the target singular diffusion equation under three distinct asymptotic regimes: the simultaneous limit, the limit via a parabolic-elliptic system, and the limit via a uniformly parabolic equation. This approach effectively transfers the diffusion singularity into the reaction terms, yielding a highly tractable system for both theoretical analysis and computation. Finally, we present numerical experiments that validate our theoretical convergence results and demonstrate the practical efficacy of the proposed approximation scheme.

Reaction-Diffusion System Approximation to the Fast Diffusion Equation

Abstract

This paper proposes a novel reaction-diffusion system approximation tailored for singular diffusion problems, typified by the fast diffusion equation. While such approximation methods have been successfully applied to degenerate parabolic equations, their extension to singular diffusion-where the diffusion coefficient diverges at low densities-has remained unexplored. To address this, we construct an approximating semilinear system characterized by a reaction relaxation parameter and a time-derivative regularizing parameter. We rigorously establish the well-posedness of this system and derive uniform a priori estimates. Using compactness arguments, we prove the convergence of the approximate solutions to the unique weak solution of the target singular diffusion equation under three distinct asymptotic regimes: the simultaneous limit, the limit via a parabolic-elliptic system, and the limit via a uniformly parabolic equation. This approach effectively transfers the diffusion singularity into the reaction terms, yielding a highly tractable system for both theoretical analysis and computation. Finally, we present numerical experiments that validate our theoretical convergence results and demonstrate the practical efficacy of the proposed approximation scheme.

Paper Structure

This paper contains 18 sections, 7 theorems, 102 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Formulation of the problems and main results
  3. Revisiting the approximation for degenerate parabolic equation
  4. Construction of the approximation for singular diffusion equation
  5. Assumptions and main results
  6. A priori Estimates and Proof of Main Theorems
  7. Key structural relations and auxiliary functions
  8. Unique existence of solutions to the approximate problems
  9. Uniqueness of the weak solution of $\mathrm{(P)}$
  10. Uniform estimates
  11. Simultaneous and parabolic-elliptic convergence
  12. Convergence of relaxation
  13. Convergence via $(P_{0,\xi})$
  14. Numerical experiments
  15. Numerical scheme
  16. ...and 3 more sections

Key Result

Theorem 2.3

Suppose that Assumptions $\mathrm{(H1)}$--$\mathrm{(H3)}$ hold. Then, both in the simultaneous limit as $\varepsilon \to 0$ and $\xi \to 0$, and in the limit as $\varepsilon \to 0$ with a fixed $\xi = 0$, the approximate solutions converge in the following sense: Here, $z$ is the unique weak solution of Problem $\mathrm{(P)}$.

Figures (4)

  • Figure 1: Initial datum $z_0$ in one and two spatial dimensions. The initial data $u_0$ and $v_0$ for \ref{['eq:scheme']} are taken accordingly to \ref{['eq:data_ini_u0']},\ref{['eq:data_ini_v0']}. $z_0$ is the unique positive solution to \ref{['eq:ini_data']} with normalization $\|z_0\|_{L^q(\Omega)}=1$.
  • Figure 2: Convergence for $\varepsilon\to 0$ and $\xi=0$ (Figure \ref{['subfig:1d_xi_0']}) or $\xi=\varepsilon$ (Figure \ref{['subfig:1d_xi_eps']}). $\mu=0.5$, $q=2.5$, $T=0.6$, $\Omega = (0,1)$, $\Delta t = 10^{-4}$, $\Delta x = 10^{-2}$.
  • Figure 3: Convergence for $\varepsilon\to 0$ and $\xi=0$ (Figure \ref{['subfig:1d_xi_0_2d']}) or $\xi=\varepsilon$ (Figure \ref{['subfig:1d_xi_eps_2d']}). $\mu=0.4$, $q=2.5$, $T=0.18$, $\Omega = (0,1)^2$, $\Delta t = 10^{-4}$, $\Delta x = 10^{-2}$.
  • Figure 4: Evolution of $\|\mathbf{z}^n\|_{\ell^q_h(\Omega)}$ and $\|(z_*(x_i,t))_{i\in\mathcal{I}}\|_{\ell^q_h(\Omega)}$. $\xi=\varepsilon=10^{-4}$, $\mu=0.5$, $q=2.5$, $\Omega=(0,1)$, $\Delta t=10^{-4}$, $h=10^{-2}$, and $u_0, v_0$ defined by \ref{['eq:ini_data']}, \ref{['eq:data_ini_u0']}, \ref{['eq:data_ini_v0']}.

Theorems & Definitions (16)

  • Definition 2.1: Weak Solution of Problem $\mathrm{(P)}$
  • Theorem 2.3: Simultaneous and Parabolic-Elliptic Convergence
  • Theorem 2.4: Convergence of Relaxation
  • Remark 2.5: Convergence via $(P_{\varepsilon,0})$
  • Theorem 2.6: Convergence via $(P_{0,\xi})$
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 6 more