Reaction-Diffusion System Approximation to the Fast Diffusion Equation

1. Problem Statement

The paper addresses the mathematical and numerical challenges associated with the Fast Diffusion Equation (FDE), a singular nonlinear diffusion problem. The target equation is given by:
$$\frac{\partial}{\partial t}\alpha(z) = \Delta z$$
where $z(x,t)$ is the unknown quantity, $\Delta$ is the Laplacian, and $\alpha$ is a constitutive function.

• Singularity: Unlike the porous medium equation (degenerate diffusion), the FDE corresponds to the case where the diffusion coefficient diverges as the density approaches zero (e.g., $\beta(z) = z^m$ with $0 < m < 1$, implying $\alpha = \beta^{-1}$ has a derivative $\alpha' \to \infty$ as $z \to 0$).
• Challenges: This singularity leads to phenomena such as finite-time extinction (the solution vanishes completely in finite time) and makes standard analytical and numerical methods difficult to apply due to the lack of uniform parabolicity.
• Gap: While reaction-diffusion system approximations have been successfully developed for degenerate parabolic equations (where diffusion vanishes), their extension to singular diffusion (where diffusion blows up) had remained unexplored.

2. Methodology

The authors propose a novel reaction-diffusion system approximation that transfers the diffusion singularity into the reaction terms, converting the singular problem into a tractable semilinear system.

2.1. Construction of the Approximating System

The authors introduce a decomposition of the state variable $z$ into two components, $u$ and $v$, such that $z = \mu u + v$, where $\mu > 0$ is a fixed parameter. They propose two variants of the approximating system $(P_{\varepsilon, \xi})$ involving a reaction relaxation parameter $\varepsilon > 0$ and a time-derivative regularizing parameter $\xi \geq 0$:

1. Fully Parabolic System ($\xi > 0$):
   $$\begin{cases} \frac{1}{\mu} \frac{\partial u}{\partial t} = \Delta u - \frac{1}{\varepsilon}(u - \alpha(\mu u + v)) \\ \xi \frac{\partial v}{\partial t} = \Delta v + \frac{\mu}{\varepsilon}(u - \alpha(\mu u + v)) \end{cases}$$
   Here, both $u$ and $v$ evolve in time. The term $\frac{1}{\varepsilon}(u - \alpha(z))$ acts as a stiff reaction forcing $u$ to relax toward $\alpha(z)$.

2. Parabolic-Elliptic System ($\xi = 0$):
   $$\begin{cases} \frac{1}{\mu} \frac{\partial u}{\partial t} = \Delta u - \frac{1}{\varepsilon}(u - \alpha(\mu u + v)) \\ 0 = \Delta v + \frac{\mu}{\varepsilon}(u - \alpha(\mu u + v)) \end{cases}$$
   Here, $v$ adjusts instantaneously (elliptic constraint) to satisfy the reaction balance.

Mechanism: By summing the equations (multiplying the first by $\mu$ and adding to the second), the reaction terms cancel out, recovering the relation $\frac{\partial}{\partial t}(\mu u + \xi v) = \Delta(\mu u + v)$. In the limit $\varepsilon \to 0$, the reaction term forces $u \to \alpha(z)$, formally recovering the original singular equation.

2.2. Analytical Framework

• Assumptions: The constitutive function $\alpha$ is assumed to be non-decreasing, locally Lipschitz, and surjective. The parameter $\mu$ is chosen such that $L_\alpha < 1/\mu$, ensuring the auxiliary function $\eta(s) = s - \mu \alpha(s)$ is strictly monotone.
• Tools: The analysis relies on:
  • Yosida Approximations: To handle the multivalued nature of inverse functions in the singular limit.
  • A Priori Estimates: Deriving uniform $L^\infty$ and energy estimates independent of $\varepsilon$ and $\xi$.
  • Compactness Arguments: Using the Aubin-Lions lemma to extract convergent subsequences.
  • Monotonicity: Leveraging the maximal monotonicity of the operators to prove uniqueness and convergence.

3. Key Contributions and Results

3.1. Well-Posedness

The authors prove the existence and uniqueness of strong solutions for the approximating systems $(P_{\varepsilon, \xi})$ for both $\xi > 0$ and $\xi = 0$. They establish that the solutions remain uniformly bounded in $L^\infty$ and $H^1$ spaces, independent of the small parameters.

3.2. Convergence Theorems

The paper rigorously establishes the convergence of the approximate solutions $(u_{\varepsilon, \xi}, v_{\varepsilon, \xi})$ to the unique weak solution $z$ of the target singular problem $(P)$ under three distinct asymptotic regimes:

1. Simultaneous Limit: As $\varepsilon \to 0$ and $\xi \to 0$ simultaneously.
   • Result: $u_{\varepsilon, \xi} \to \alpha(z)$ and $z_{\varepsilon, \xi} \to z$ strongly in $L^2$.
2. Relaxation Limit (Parabolic-Elliptic): For fixed $\varepsilon > 0$, as $\xi \to 0$.
   • Result: The fully parabolic system converges to the parabolic-elliptic system.
3. Iterated Limit (Regularized FDE): For fixed $\xi > 0$, as $\varepsilon \to 0$, followed by $\xi \to 0$.
   • Result: The system first converges to an intermediate "regularized" fast diffusion equation (where the singularity is smoothed by $\xi$), and then to the original singular FDE.

3.3. Numerical Validation

The authors implemented a fully implicit finite-difference scheme for the approximating system.

• Convergence Rate: Numerical experiments in 1D and 2D (using $\alpha(s) = |s|^{q-2}s$ with $q=2.5$) demonstrated first-order convergence ($O(\varepsilon)$) of the approximate solution to the analytical solution of the FDE.
• Extinction Phenomenon: While the discrete scheme does not strictly exhibit finite-time extinction (due to the nature of implicit discretization preserving positivity), the numerical results successfully captured the rapid exponential decay of the solution norm past the theoretical extinction time $T^*$, validating the scheme's ability to model the physical behavior of the FDE.

4. Significance

• Theoretical Breakthrough: This work fills a critical gap in the theory of reaction-diffusion approximations by extending the framework from degenerate to singular diffusion. It provides a rigorous mathematical foundation for handling diffusion coefficients that diverge at low densities.
• Computational Utility: By transforming the singular nonlinear diffusion term into a reaction term, the method allows for the use of linear numerical schemes (or semi-linear systems) which are significantly easier to implement and computationally cheaper than direct methods for singular PDEs.
• Flexibility: The introduction of the parameter $\xi$ offers flexibility in modeling. Setting $\xi=0$ yields a parabolic-elliptic system (computationally efficient for steady-state-like behavior), while $\xi>0$ provides a fully parabolic system suitable for time-dependent analysis.
• Applicability: The approach is applicable to a broad class of singular problems, including models for population dynamics, filtration, and phase transitions where standard diffusion assumptions fail.

In summary, Murakawa and Salin provide a robust, theoretically sound, and numerically efficient approximation strategy for the Fast Diffusion Equation, effectively managing the singularity through a reaction-relaxation mechanism.