Nonlinear Porous Medium Equation (NPME)

• NPME is a class of degenerate nonlinear parabolic equations that incorporate nonlocal interactions through fractional operators to model anomalous diffusion phenomena.
• The equation employs a fractional pressure law and self-similar Barenblatt solutions to describe finite-speed propagation and long-range effects in porous media.
• Robust numerical and analytical methods, including energy estimates and Galerkin approximations, support existence, regularity, and inverse problems for NPMEs.

The nonlinear porous medium equation (NPME) is a class of degenerate nonlinear parabolic equations describing diffusive processes in media with nonlinear pressure laws and frequently incorporating nonlocal effects. The NPME generalizes the classical porous medium equation by allowing both nonlinear and spatially nonlocal fluxes, which are crucial for modeling anomalous transport phenomena, such as superdiffusive moisture evolution, long-range particle jumps, or nonlocal mechanical interactions in porous substrates.

1. Model Formulation and Nonlocal Pressure Law

A prototypical NPME studied in the literature assumes the form

$$u_t = \nabla \cdot \left( u^{m-1} \nabla p \right), \quad p = (-\Delta)^{-s} u, \quad x \in \mathbb{R}^N, \ t > 0,$$

with initial data

$u(x,0) = u_0(x) \geq 0,$

where $m>1$ denotes the nonlinear degeneracy exponent and $s\in(0,1)$ is the order of the fractional potential defining the nonlocality in pressure. The operator %%%%2%%%% is realized as the Riesz potential

$$(-\Delta)^{-s}u(x) = C_{N,s} \int_{\mathbb{R}^N} |x-y|^{-(N-2s)} u(y) \, dy,$$

and $p$ solves $(-\Delta)^s p = u$. The flux $\nabla p$ introduces long-range interactions in the pressure field, distinguishing nonlocal NPMEs from their classical (local) counterparts (Stan et al., 2018, Płociniczak, 2018).

Alternative formulations employ a fractional gradient flux, notably

$$u_t = \nabla \cdot \left( u^m \nabla^{\alpha-1} u \right), \quad 0<\alpha<2,$$

preferred in hydrological modeling for yielding a pressure law $p(u) = u^{m-1}$ and a flux interpretation tied directly to capillary pressure via Darcy's law (Płociniczak, 2018).

In bounded domains, the nonlocal operator may be realized spectrally, with $p = (-\Delta)^{-s} (\vert u \vert^{m_2-1}u)$, leveraging Dirichlet eigenfunction expansions to encode boundary conditions (Nguyen et al., 2017).

2. Existence, Uniqueness, and Regularity Theory

Global existence of nonnegative weak solutions is established for nonlocal NPMEs with $u_0$ a finite nonnegative Radon measure, or $u_0 \in L^1 \cap L^\infty$ (Stan et al., 2018, Caffarelli et al., 2010):

• Mass conservation holds: $\int_{\mathbb{R}^N} u(x,t)\, dx = \int u_0$.
• Smoothing effects: $L^\infty$ norms decay algebraically,

$$\|u(\cdot,t)\|_\infty \leq Ct^{-\alpha} (\int u_0)^{\beta},$$

with explicit exponents $\alpha$, $\beta$ depending on $N$, $m$, and $s$.

• Energy estimates are available for $L^p$ norms and nonlocal Sobolev norms, reflecting regularization through nonlocal pressure flow.

Uniqueness is obtained in one space dimension ($N=1$) via quasilinear nonlocal comparison principles for the integrated variable $v(x,t) = \int_{-\infty}^x u(y,t) dy$ (Stan et al., 2018). For $N>1$, uniqueness remains open due to the lack of a general comparison principle; existence results extend to bounded Lipschitz domains using Galerkin approximations and monotonicity arguments (Lin et al., 2023, Nguyen et al., 2017).

Regularity theory is significantly affected by nonlocality:

• For linear fractional heat flows ($m=1$), solutions exhibit instant regularization (Płociniczak, 2018).
• For $m>1$, finite speed of propagation is retained for the pressure form, contrasting with infinite speed for fractional Laplacian evolution of $u^m$ (Płociniczak, 2018, Caffarelli et al., 2010).
• Hölder continuity in space-time can be established under mild data assumptions (Płociniczak, 2018).
• Sharp $C^{2,\alpha}$ regularity estimates for fully nonlinear porous medium-type equations have been proven in bounded domains, using transformations reducing the degenerate prefactor to a non-degenerate form amenable to parabolic Schauder theory (Yun, 29 Jul 2024).

3. Self-Similar Solutions and Asymptotic Behavior

A central focus in the theory is the identification and analysis of self-similar Barenblatt-type solutions. These take the canonical scaling form

$$u(x, t) = t^{-\alpha} F\left( x t^{-\beta} \right),$$

where

$$\alpha = \frac{N}{N(m-1) + 2 - 2s}, \quad \beta = \frac{1}{N(m-1) + 2 - 2s},$$

with $F(\xi)$ solving an integro-differential profile equation

$$\nabla \cdot [F^{m-1} \nabla (-\Delta)^{-s} F] + \alpha F + \beta \xi \cdot \nabla F = 0.$$

• For $m < 2$, these profiles are obtained via algebraic transformation from fractional porous medium profiles.
• For $N=1$, $m \ge 2$, existence follows from compactness arguments on suitable rescalings (Stan et al., 2018).
• Asymptotic behavior in $N=1$ is characterized by convergence in $L^p$ norms to the unique self-similar solution of mass $M$ (Stan et al., 2018).

Explicit formulas for Barenblatt solutions with compact support have been derived for various choices of nonlocality order $\alpha$ and exponent $m$, notably in the works of Biler–Imbert–Karch (Biler et al., 2013), De Gregorio (Gregorio, 2019), and are tied to probabilistic representations using random flight processes.

4. Numerical Schemes and Computational Aspects

Numerical approximation of NPMEs poses particular challenges owing to nonlocality and degeneracy. Energetic variational approaches provide robust schemes, guaranteeing energy stability and unique solvability. For the classical PME in Eulerian and Lagrangian forms:

• Second-order accurate schemes in both time and space have been developed, employing modified Crank–Nicolson methods and discrete variational principles.
• Convexity arguments ensure unique minimizers for the discrete system; discrete energy dissipation holds for the scheme.
• Higher-order asymptotic expansions and two-step error analysis yield optimal convergence rates $O(\tau^2 + h^2)$ in both $L^2$ and $L^\infty$ norms (Duan et al., 2020).

5. Inverse Problems, Identification, and Stochastic Representations

Recent advances address inverse problems for NPMEs, concerning the unique recovery of coefficients and kernels from boundary measurements.

• Under suitable regularity and ellipticity assumptions, the Dirichlet-to-Neumann map associated with the nonlocal operator uniquely determines density, absorption, and conductivity kernel (Lin et al., 2023).
• Probabilistic representations link explicit Barenblatt-type solutions to random flight models, with the law of suitably time-rescaled isotropic transport processes matching the fundamental solution of the NPME. This correspondence rigorously connects the macroscopic PDE with underlying stochastic microscopic dynamics (Gregorio, 2019).

6. Comparisons with Other Diffusive Models and Extensions

The NPME framework interpolates between classical PME and fractional porous medium models:

• Standard PME, $u_t = \Delta(u^m)$, possesses finite speed of propagation, local pressure, and explicit Barenblatt solution.
• Fractional PME, $u_t + (-\Delta)^\sigma(u^q) = 0$, exhibits infinite speed of propagation but maintains mass-conserving self-similar profiles.
• NPME recovers linear fractional heat flow for $m = 1$; for $m = 2$, finite speed of propagation prevails, as first studied by Caffarelli–Vázquez (Stan et al., 2018).
• Nonlocal forms with fractional time derivatives further generalize the NPME, with Crandall–Liggett theory and De Giorgi–Moser methods providing existence, uniqueness, and Hölder continuity (Djida et al., 2018).

Open problems remain in establishing uniqueness for $N > 1$, regularity of weak solutions, detailed numerical analysis, and extension to more general nonlocal pressure laws and coupled systems, including chemotactic drift (Stan et al., 2018, Caffarelli et al., 2010, Yun, 29 Jul 2024).

7. Applications and Broader Context

The NPME is relevant in modeling:

• Hydrological superdiffusion and anomalous moisture transport in media with heterogeneous or networked pore structures (Płociniczak, 2018).
• Population flows and tumor growth with mechanical constraints, especially in incompressible limits transitioning to free boundary problems (Dalibard et al., 2021).
• Random flights and anomalous heat transport, where finite speed and spatial intermittency are fundamental (Gregorio, 2019).
• Graph-based diffusion processes, with curvature–dimension inequalities yielding discrete analogues of continuous kernel and gradient bounds (Man, 2019).
• Nonlocal mechanics, pattern formation, and geophysical models where long-range interactions cannot be ignored.

The mathematical theory of NPMEs synthesizes partial differential equations, potential theory, stochastic analysis, and geometric methods, forming a rich research interface between analysis and applied modeling.