Nonlocal Advection-Diffusion Equations

• Nonlocal advection-diffusion equations are PDEs featuring integral operators that capture long-range interactions and anomalous transport dynamics.
• They model phenomena such as biological aggregation, geophysical flows, and Lévy flights by coupling nonlocal advection with fractional diffusion.
• Numerical strategies, including operator splitting and PINNs, as well as gradient flow formulations, effectively address the challenges of nonlocality and singularities.

Nonlocal advection-diffusion equations are a fundamental class of partial differential equations (PDEs) in which transport processes, diffusive mechanisms, and possibly reactive terms interact via nonlocal operators. The “nonlocal” aspect means that the spatial operators involve convolution or integral terms—often reflecting underlying jump dynamics, long-range interactions, or finite-range sensing—rather than purely differential expressions. These equations capture a rich variety of phenomena in physics, biology, and applied mathematics, including anomalous diffusion, biological aggregation, geophysical flows, and more. They also present substantial mathematical and computational challenges due to their nonlocality, possible singularities, and complex coupling of nonlinear advection with nonlocal diffusion.

1. Fundamental Structure and Mathematical Formulation

A broad class of nonlocal advection-diffusion models has the form: $u_t + \nabla \cdot (u\, v(u)) = \mathcal{A}(u)$ where:

• $u(x,t)$ is the density or concentration.
• The advection velocity $v(u)$ may depend nonlocally on $u$, e.g., $v(u)(x) = \int_{\Omega} \psi(x-y) u(y) dy$ for some kernel $\psi$.
• The diffusion (or dispersion) operator $\mathcal{A}$ is linear but may be nonlocal, e.g., a (possibly fractional or anisotropic) Laplacian.

Canonical operators and examples:

• Fractional Laplacian: $\mathcal{A}(u) = -(-\Delta)^{\alpha/2} u$, with Fourier symbol $-|\xi|^\alpha$ for $\alpha \in (0,2)$, modeling Lévy flights and superdiffusion (Deng et al., 2018).
• Anisotropic/tempered extensions: E.g., operators of the form

$$\Delta_m^{\beta/2} p(x) = \frac{1}{|\Gamma(-\beta)|} \int_{\mathbb{R}^n} [p(x-y) - p(x) + \chi_{|y|<1} (y \cdot \nabla p(x)) ]\, \frac{m(y)}{|y|^{n+\beta}}\, dy$$

where $m(y)$ prescribes directional weights.

• Integral nonlocal diffusion: $\mathcal{A}(u)(x) = \int_Q W(x,y) D(u(y)-u(x)) dy$, for $D$ Lipschitz (Kaliuzhnyi-Verbovetskyi et al., 2019).

Advection can itself be nonlocal or nonlinear, as in models for aggregation and cell movement: $v(u) = K * u$ with an interaction kernel $K$ (possibly singular), or velocity fields determined by singular integral operators in geophysical flows (Holden et al., 2012).

Boundary and initial conditions are set either locally (standard Dirichlet/Neumann), nonlocally (specifying averages or fluxes on a subset), or on the complement of the domain, reflecting the global influence of jumps or interactions (Deng et al., 2018Miller et al., 2017).

2. Analytical Properties: Well-Posedness, Regularity, and Existence Results

Mathematical analysis of nonlocal advection-diffusion equations hinges on several features:

• Well-posedness is established under assumptions of regularity and nondegeneracy for underlying kernels (e.g., support of $m(y)$ spans $\mathbb{R}^n$ (Deng et al., 2018)).
• For active scalar equations with nonlinear advection and nonlocal diffusion, solutions are globally well-posed for sufficiently regular initial data (e.g., $u_0 \in H^k$, $k \geq 6$) (Holden et al., 2012).
• In multi-species models with nonlocal kernels (possibly distinct per species), global existence and positivity of weak solutions hold in arbitrary dimension, and classical positive solutions are shown in 1D via comparison and Harnack inequalities (Giunta et al., 2023Giunta et al., 2021Cozzi et al., 11 Mar 2024).
• The role of nonlocality: Nonlocal interaction regularizes the dynamics, preventing blow-up scenarios that appear in the local limit (e.g., chemotaxis may blow up in finite time locally, but the nonlocal version remains globally bounded) (Giunta et al., 2023).
• In degenerate, anisotropic settings, global existence of very weak solutions is obtained provided the diffusion degeneracy set has upper box fractal dimension below a threshold, ensuring diffusion dominates on a “large enough” part of the domain (Eckardt et al., 21 Jun 2024).

Fractional time derivatives and nonlocality in time are also considered, leading to equations with long-range memory and subdiffusive dynamics. Regularity theory in these settings requires energy estimates involving fractional integrals and new Grönwall-type inequalities (McLean et al., 2019).

3. Numerical Schemes and Implementation Strategies

Numerical solution of nonlocal advection-diffusion equations demands special care due to the integral nature and possible singularities:

• Operator splitting methods (Godunov, Strang) used for equations with both nonlinear advection and (possibly nonlocal) diffusion/dispersion yield provable convergence rates: first order for Godunov, second order for Strang, under high Sobolev regularity (Holden et al., 2012). These techniques allow hyperbolic and diffusive steps to be addressed by tailored solvers.
• Probabilistic approaches: Discrete time random walk (DTRW) schemes are derived as continuum limits of properly constructed lattice jump processes. The stochastic process enforces nonnegativity, mass conservation, and delivers robust stability and accurate shock/discontinuity resolution (Angstmann et al., 2016).
• Sparse Monte Carlo and Galerkin techniques: Nonlocal terms may be efficiently approximated via sparse Monte Carlo quadrature coupled with discontinuous Galerkin projections, ensuring convergence even when the kernel is singular or discontinuous. Convergence rates depend explicitly on the sparsity and kernel regularity (Kaliuzhnyi-Verbovetskyi et al., 2019).
• Spectral methods: For periodic domains, spectral (FFT-based) methods are effective, particularly in multi-species models, as convolution and differentiation translate to low-cost algebraic operations (Giunta et al., 2021).
• Physics-informed neural networks (PINNs): Recent progress leverages the Schwarz waveform relaxation method with PINNs to parallelize and adaptively train domain-decomposed surrogate models capable of handling both local and nonlocal operators (Lorin et al., 2021). PINN-based schemes allow flexible architecture adaption in subdomains dependent on local solution complexity.

4. Model Classes, Applications, and Phenomenology

The literature encompasses a wide spectrum of model classes and applications:

• Aggregation–diffusion and biological movement: Nonlocal advection terms model sensing and movement in cells and animals, with aggregation driven by nonlocal averages (e.g., convolution with $K$) and diffusion regularizing against blow-up (Painter et al., 2023Giunta et al., 2021Giunta et al., 2023Cozzi et al., 11 Mar 2024). Multi-species interactions—attractive or repulsive, with distinct kernels—yield pattern formation, segregation, and oscillatory or traveling waves.
• Pattern formation and bifurcation: Linear stability and bifurcation analyses reveal that symmetric kernels tend to generate stationary (Turing) patterns, while asymmetric (advection-induced) kernels drive oscillatory instabilities and propagating waves (Siebert et al., 2014). The order and width of the nonlocal kernel control the selection and robustness of emergent spatial structures.
• Nonlocal-to-local limits: Taylor expansions of the convolution operator in the regime of short-range interactions lead to higher-order local PDEs (e.g., thin-film or Cahn–Hilliard type), preserving key phenomenology but simplifying implementation and analysis (Falcó et al., 13 May 2025). Such limits clarify the connection between nonlocal aggregation models and surface-tension-driven sorting.
• Nonlocal Fokker–Planck and fractional diffusion: Fractional and tempered Laplacians account for heavy-tailed jump processes, with stochastic integral formulations and connections to anomalous diffusion, Lévy flights, and random media (Deng et al., 2018Warren, 30 Dec 2024Choi et al., 20 May 2024). Nonlocal Fokker–Planck models can be naturally recast as gradient flows in nonlocal Wasserstein-type metrics.

5. Gradient Flow Structure and Entropy Methods

A powerful unifying perspective frames many nonlocal advection-diffusion equations as gradient flows of suitable energy functionals in measure spaces endowed with nonlocal (possibly fractional or anisotropic) Wasserstein or entropy metrics:

• For equations of the form

$$\partial_t \rho_t + \int_X \left[ \frac{d\rho_t}{d\pi}(x) - \frac{d\rho_t}{d\pi}(y) \right] \eta(x, y) d\pi(y) = 0$$

the dynamics are the gradient flow of the relative entropy $\mathcal{H}(\rho|\pi)$ in a metric space whose distance is defined by a nonlocal action functional (Warren, 30 Dec 2024).

• This “nonlocal Otto calculus” generalizes the classical Wasserstein-2 metric (for local Fokker–Planck) to settings with general jump kernels; curves of maximal slope correspond to entropy solutions, and exponential convergence to equilibrium holds under nonlocal log–Sobolev inequalities.
• Similar gradient flow formulations underlie the Jordan–Kinderlehrer–Otto (JKO) scheme for constructing weak measure-valued solutions to nonlocal (possibly degenerate) diffusion equations, with particle approximations arising from Lagrangian ODEs for the empirical measure (Carrillo et al., 2023).
• Coupling local and nonlocal evolution via an energy functional enables mixed models, preserving mass and long-term decay properties. Moreover, the rescaling of the nonlocal kernel provides a rigorous bridge to classical local advection-diffusion models (Gárriz et al., 2019Falcó et al., 13 May 2025).

6. Challenges, Future Directions, and Open Problems

Although significant progress has been made across theory, numerics, and modeling, several challenges and research directions remain:

• Local-to-nonlocal limits: Ensuring continuity of solution behavior under vanishing interaction length and clarifying connections between nonlocal and higher-order local models, including regularity and minimizer properties for nonconvex energies (Falcó et al., 13 May 2025).
• Blow-up vs. regularization: Rigorous quantification of the blow-up threshold as the nonlocality parameter tends to zero; precise characterization of dimensions and kernel regularity preserving or destroying global existence (Giunta et al., 2023Cozzi et al., 11 Mar 2024Eckardt et al., 21 Jun 2024).
• Numerical analysis: Development of computationally efficient, low-regularity-compatible methods that guarantee convergence and preserve qualitative solution properties in the presence of singularities, anisotropy, or stochastic noise (Kaliuzhnyi-Verbovetskyi et al., 2019Choi et al., 20 May 2024).
• Data-driven calibration: Systematic inference and parameter estimation in nonlocal models to match experimental or observational data, leveraging the simplified form of local models derived in short-range limits (Falcó et al., 13 May 2025).
• Stochastic nonlocal equations: Regularity and pathwise analysis for SPDEs with nonlocal (variable-order) diffusion operators and spatially colored noise, under reinforced integrability (Dalang-type) conditions (Choi et al., 20 May 2024).

7. Summary Table: Model Classes and Central Features

Model/Class	Nonlocal Advection	Nonlocal Diffusion	Key Applications/Features
Active scalar (SQG, aggregation)	$v(u) = \psi * u$ or curl	Fractional Laplacian	Fluid dynamics, biological aggregation
Multi-species models	Distinct $K_{ij}$ per pair	Diffusion + nonlocal drag	Pattern formation, segregation, oscillation
Anomalous/fractional dynamics	(may be absent)	$(-\Delta)^{\alpha/2}$	Lévy flights, anomalous transport
Coupled local/nonlocal regions	Drift $\nabla V(x)$	Laplacian + integral	Domains with heterogeneous diffusive behavior
Cahn–Hilliard/Thin-film limit	Derived from nonlocal	Fourth-order local PDE	Cell sorting, surface tension
Stochastic equations	Variable/coefficient $b$	$\phi(\Delta)$ nonlocal	SPDEs, random media

The synthesis of analytical, computational, and modeling advances in nonlocal advection-diffusion equations underpins much of the current understanding of collective dynamics in complex, spatially-extended systems. Nonlocality both enriches the range of possible behaviors—from self-organizing patterns to anomalous transport and robust regularization—and presents new avenues for theoretical and applied research.