Nonlocal Conservation Laws: Theory, Numerics, Applications

• Nonlocal conservation laws are PDEs where the flux at each point depends on the local value and an aggregated spatial average via convolution.
• They exhibit unique properties such as the failure of the maximum principle, dramatic total variation growth, and infinite-speed information propagation.
• Their applications span traffic flow, crowd dynamics, and wave mechanics, spurring advances in both numerical methods and theoretical analysis.

A nonlocal conservation law is a partial differential equation in which the conserved flux at each point depends not only on the local value of the conserved variable but also on an aggregate or "averaged" value over a spatial region, typically encoded by convolution with a spatial kernel. These models generalize classical (local) conservation laws—ubiquitous in fluid dynamics, traffic flow, and continuum mechanics—by directly incorporating long-range interactions, memory effects, or distributed influences. The inclusion of nonlocality dramatically alters the analytical, numerical, and structural properties of the PDE, leading to new phenomena and raising fresh challenges for both theory and computation.

1. Formulation and Examples of Nonlocal Conservation Laws

A canonical scalar nonlocal conservation law in one dimension takes the form

$$\partial_t \rho(t,x) + \partial_x\big[ f\big(t,x,\rho(t,x)\big) \, v\big((\rho(t,\cdot) * \eta)(x)\big) \big] = 0,$$

where the nonlocality enters via convolution,

$$(\rho * \eta)(x) = \int_{\mathbb{R}} \rho(\xi) \, \eta(x-\xi)\,d\xi.$$

Here, $\eta$ is a (possibly anisotropic or compactly-supported) kernel, and $v$ may be nonlinear. The local conservation law is recovered in the singular limit when $\eta\to\delta$ (the Dirac measure), yielding $$\partial_t\rho + \partial_x(f(t,x,\rho) v(\rho)) = 0$$ (Amorim et al., 2013).

Nonlocal models with physical or applied relevance include:

• Traffic flow: Nonlocal terms represent the "look-ahead" effect, with kernels supported on $(-\infty,0]$ so that drivers only sense downstream density (Colombo et al., 2023).
• Peridynamics: Nonlocal elastic interactions with finite horizon.
• Crowd and pedestrian dynamics: Nonlocal operators encode the crowd's perception of global or neighborhood averages (Rossi et al., 2017).
• Wave mechanics: Nonlocal Lagrangians generalize conservation principles (Spourdalakis et al., 2015).

Several models allow for both nonlocal dependence in the density and the velocity: $$\partial_t q + \partial_x\big( V_1( \gamma * V_2(q) ) \, q \big) = 0,$$ with various choices of $V_1$, $V_2$ and convolution kernel $\gamma$ (Friedrich et al., 2023).

2. Analytical Properties and Theoretical Distinctions

Nonlocality induces several pronounced differences relative to local conservation laws:

• Maximum Principle Failure: Solutions to classical scalar conservation laws preserve bounds on the initial data (maximum principle). Nonlocal fluxes may break this, allowing overshoot/undershoot beyond the initial profile (Amorim et al., 2013).
• Growth of Total Variation: In local laws, the total variation (TV) is nonincreasing for entropy solutions. Nonlocal terms can produce sharp (even explosive) TV increases, leading to more oscillatory dynamics (Amorim et al., 2013, Colombo et al., 5 Aug 2024). For certain monotonicity-preserving nonlocal models, TV bounds can be maintained under additional structural assumptions (Du et al., 2016).
• Propagation Speed: The convolution kernel typically induces infinite-speed information propagation (strict hyperbolicity is lost), though in discretizations the CFL condition still controls numerical stability (Amorim et al., 2013).
• Finite-Time Blowup: For kernels with weak regularity (BV or Sobolev), solutions may undergo finite-time blowup, concentrating into measures (Colombo et al., 5 Aug 2024).
• Nonuniqueness After Blowup: Different smooth approximations to a non-smooth kernel may yield different weak* limits after blowup, preventing canonical extension of solutions (Colombo et al., 5 Aug 2024).

3. Numerical Schemes and Convergence Analysis

Numerically, nonlocal conservation laws are commonly discretized using finite volume schemes adapted from classical methods, with special care to maintain monotonicity and control the convolution term. The following strategies are prominent:

• Lax–Friedrichs–type Schemes: Incorporate numerical diffusion, discretizing both local flux and the nonlocal convolution over a uniform mesh. Positivity, $L^1$-mass conservation, $L^\infty$-bounds, and a priori estimates on TV and time-regularity are crucial to guarantee convergence to a weak (entropy) solution. Discrete entropy inequalities (Kruzhkov-type) are established in the presence of nonlocality (Amorim et al., 2013).
• Second-order High-Resolution Schemes: The MUSCL–Hancock framework can be adapted to nonlocal equations by designing a locally second-order convolution quadrature and employing mesh-dependent modifications to standard limiters (e.g., minmod with mesh-based threshold) to secure entropy admissibility and suppress spurious high-order corrections. TV and $L^1$–Lipschitz estimates, along with compactness (e.g., Helly’s theorem), underpin convergence (Manoj et al., 4 Jun 2025).
• Asymptotic Compatibility and Nonlocal CFL Conditions: Discretizations must ensure that as the horizon parameter for nonlocality shrinks (e.g., via the kernel width), solutions of the nonlocal scheme converge to solutions of the local law. Appropriate nonlocal CFL conditions and monotone fluxes guarantee that the numerical approach respects both the nonlocal-to-local limit and entropy selection (Du et al., 2016).

A comparison of scheme features is provided below.

Scheme Type	Nonlocal Term Treatment	Entropy Consistency
Lax–Friedrichs	Nodewise discrete convolution	Discrete Kruzhkov entropy
MUSCL–Hancock (MH)	Piecewise linear reconstruct., quadrature	Mesh-dependent limiter for entropy
Godunov, FV	Monotone, flux splitting over stencils	Discrete entropy & max principle

4. Nonlocal-to-Local Limit and Compactness Results

A central analytical issue is characterizing the convergence as the kernel becomes sharply localized (i.e., $\eta_\epsilon \rightarrow \delta$ as $\epsilon \to 0$). Rigorous results employ:

• Uniform Total Variation or BV estimates: For exponential-type kernels, the total variation of the nonlocal average (e.g., $W_\eta[q]$ or $(q*\eta_\epsilon)$) is uniformly bounded in $\epsilon$ (Coclite et al., 2020, Coclite et al., 2023, Colombo et al., 2023), ensuring compactness and allowing strong $L^1_{\text{loc}}$ convergence.
• Oleĭnik-type Entropy Conditions: Nonlocal analogues of the one-sided Lipschitz (Oleĭnik) estimate provide crucial BV-regularization, precluding non-entropy shocks and ensuring the selection of the physically relevant limit solution without requiring $BV$ initial data (Coclite et al., 2023).
• Monotonicity Preservation: For certain models (“pair-interaction” or “nonlocal-in-velocity”), monotonic initial data remain monotonic for all time, directly aiding compactness and entropy admissibility (Friedrich et al., 2022, Du et al., 2016).
• Singular Limit Results for Unsigned Data: Nonlocal approximations based on averages of $|q|$ yield convergence to local entropy solutions even without a sign constraint on initial data, extending previous sign-restricted theories (Keimer et al., 2023).

Convergence to the unique entropy solution is typically ensured for a class of kernels and velocities, but pathological cases (e.g., non-BV kernels) may lead to nonuniqueness in the limit after blowup (Colombo et al., 5 Aug 2024).

5. Applications and Model Classes

Nonlocal conservation laws find applications across multiple disciplines, each harnessing the flexibility of the nonlocal flux formulation:

• Traffic Flow: Nonlocal velocities model drivers’ sensitivity to downstream traffic, accurately predicting jam formation, queue dynamics, and under-/overestimation effects (Friedrich et al., 2023, Friedrich et al., 2022, Colombo et al., 2023).
• Crowd Dynamics: Nonlocal perception in bounded domains, with boundary-adapted operators, yields models capturing lane formation, congestion, and evacuation in realistic geometries (Rossi et al., 2017).
• Wave Mechanics and Nonlocal Field Theories: Nonlocal Lagrangians generate extended families of invariants, leading to generalized conservation laws that supplement or extend classical Noether theory (Spourdalakis et al., 2015, Kegeles et al., 2015).
• Integrable Systems: Nonlocal conservation laws arise naturally via differential coverings, zero curvature representations, or Bäcklund transformations, enriching the algebraic and geometric structure of integrable PDEs (Lou, 2014, Hlaváč et al., 2016, Krasil'shchik, 2020, Morozov, 30 Jul 2025).

Specific models are engineered for anisotropic settings (e.g., upstream- or downstream-biased kernels) and can accommodate multiple interacting populations, as in vehicular or pedestrian systems.

6. Geometric and Algebraic Generalizations

Beyond scalar conservation, nonlocal conservation laws are systematically classified and constructed using geometric machinery:

• Reciprocal Transformations and Bäcklund Relations: In integrable evolutionary PDEs, nonlocal conservation laws can generate new integrable models under reciprocal transformations and, reciprocally, relate different solutions via implicit auto-Bäcklund transformations (Lou, 2014).
• Differential Coverings and Abelian Coverings: Many integrable systems admit an "abelian covering," a hierarchy of nonlocal conservation laws constructed via expansions of a Lax pair or a zero curvature representation. In the constant astigmatism equation, the covering is closed under reciprocal transformations, with functional dependencies encoded via Wronskian-type relations (Hlaváč et al., 2016, Krasil'shchik, 2020).
• Nonlocal Field Theories and Generalized Noether Theorems: In nonlocal actions (where the Lagrangian depends on several space-time arguments), the conserved currents acquire explicit nonlocal correction terms. The generalization of Noether’s theorem systematically accounts for these, providing a geometric and combinatorial foundation for conservation principles in settings such as group field theories for quantum gravity (Kegeles et al., 2015).

A notable feature is that symmetries of field correlation functions or automorphisms (affine transformations) generate entire continua of nonlocal conservation laws, as opposed to the finite family produced by local symmetries (Chafin, 2014).

7. Open Problems and Ongoing Challenges

Current research is focused on several outstanding challenges:

• Beyond Smooth Kernels: The breakdown of uniqueness and measure-valued blowup for weakly regular kernels indicates the necessity of refined solution concepts or selection criteria in the nonlocal regime (Colombo et al., 5 Aug 2024).
• Entropy Selection and Weak Regularity: Establishing entropy admissibility for broad classes of nonlocal laws (e.g., those involving unsigned data or measure-valued solutions) requires further development of analytical tools, such as nonlocal entropy inequalities, Oleĭnik-type regularity, and compactness frameworks for weakly regular kernels (Coclite et al., 2023, Keimer et al., 2023).
• Multidimensional Extension and Anisotropy: The nonlocal-to-local limit, especially for anisotropic or directionally biased kernels in higher dimensions, remains an active domain, with applications in traffic, human crowds, and granular media (Colombo et al., 2023, Colombo et al., 5 Aug 2024).
• Efficient High-Order Numerical Schemes: The development, analysis, and implementation of numerical methods that maintain high-order accuracy and entropy consistency for nonlocal PDEs—especially in multi-D or complex geometries—continues to present substantial computational and theoretical obstacles (Manoj et al., 4 Jun 2025).

The field of nonlocal conservation laws is thus distinguished by its wide applicability, the rich new phenomena induced by nonlocality, and ongoing challenges at the intersection of analysis, geometry, and numerical mathematics.