Systems of Nonlocal Conservation Laws with Memory and Their Zero Retention Limit

1. Problem Statement

The paper addresses the mathematical analysis and numerical approximation of systems of nonlocal hyperbolic conservation laws that incorporate memory effects (nonlocality in both space and time).

The governing system for $N$ coupled equations is given by:
$$\partial_t U^k + \partial_x \left( f^k(U^k) \nu^k((\Theta \ast U)_k) \right) = 0, \quad (t, x) \in (0, T) \times \mathbb{R}$$
where:

• $U = (U^1, \dots, U^N)$ is the vector of unknowns.
• The flux depends on a nonlocal operator $(\Theta \ast U)_k$, which is a convolution involving both a spatial kernel $\mu$ and a temporal memory kernel $\Gamma$.
• Specifically, $(\Theta \ast U)_k(t, x) = \sum_{j} \int_0^t \int_{\mathbb{R}} U^j(\tau, \xi) \mu_{j,k}(x-\xi) \Gamma_{j,k}(t-\tau) \, d\xi \, d\tau$.

Key Challenges:

1. Lack of Semigroup Property: The temporal convolution destroys the standard semigroup property found in local conservation laws, rendering classical well-posedness techniques inapplicable.
2. Discontinuous Flux: The temporal convolution term often lacks spatial smoothness, making the flux discontinuous with respect to the spatial variable $x$.
3. Asymptotic Limits: There is a need to understand the behavior of solutions as the memory kernel shrinks (the "memory-to-memoryless" limit) and whether numerical schemes remain consistent with this limit (asymptotic compatibility).

2. Methodology

The authors employ a combination of analytical techniques and finite volume numerical methods:

• Entropy Solution Framework: They define an entropy solution concept adapted for nonlocal systems to ensure uniqueness, as weak solutions are not unique due to the nonlinearity.
• Finite Volume Approximation: A first-order convergent finite volume scheme is proposed. The scheme uses a marching formula to update cell averages, incorporating the nonlocal convolution term via discrete approximations of the kernels.
• Stability and BV Bounds: The authors derive uniform Bounded Variation (BV) estimates for the numerical solutions. Crucially, they prove these bounds are independent of the $L^\infty$ norm of the temporal kernel $\Gamma$, depending only on its $L^1$ norm. This is essential for the asymptotic analysis.
• Relative Entropy Estimates: To analyze the convergence as the memory kernel shrinks, they utilize a relative entropy estimate (Kruzhkov-type doubling of variables) adapted for nonlocal conservation laws.
• Asymptotic Analysis: They investigate the limit as the support of the temporal kernel $\Gamma_\delta$ shrinks to a Dirac delta (i.e., $\delta \to 0$), effectively removing the memory effect.

3. Key Contributions

A. Well-Posedness Theory

• Existence and Uniqueness: The paper establishes the existence and uniqueness of entropy solutions for the coupled system with general nonlinear fluxes and general smooth kernels.
• Continuous Dependence: They prove continuous dependence on initial data, showing that the solution operator is Lipschitz continuous in the $L^1$ norm.
• Generalization: Unlike previous works restricted to specific kernels or linear fluxes, this framework handles general $N$-coupled systems with nonlinear fluxes $f^k$ and general smooth kernels.

B. Asymptotic Analysis (Memory-to-Memoryless Limit)

• Convergence to Nonlocal-Space Limit: The authors prove that as the temporal kernel radius $\delta \to 0$, the entropy solution of the space-time nonlocal system converges to the entropy solution of the purely spatially nonlocal system (where memory is removed).
• Convergence Rate: They derive an explicit error estimate of order $O(\sqrt{\delta})$ for the convergence of the exact solutions.

C. Asymptotic Compatibility of Numerical Schemes

• Scheme Compatibility: The paper proves that the proposed finite volume scheme is asymptotically compatible. This means that as both the mesh size $\Delta x \to 0$ and the memory radius $\delta \to 0$, the numerical solution converges to the entropy solution of the memoryless (purely spatial) limit.
• Error Estimates: The total error for the numerical scheme in the memoryless limit is estimated as $O(\sqrt{\delta} + \sqrt{\Delta x})$.
• Novelty: This is the first work to provide such convergence and asymptotic analysis for finite volume schemes applied to nonlocal conservation laws with memory.

D. Relaxation of Geometric Constraints

• The analysis is conducted without imposing geometric restrictions (such as convexity) on the spatial or temporal convolution kernels, which is a significant improvement over existing literature on nonlocal-in-space-only laws.

4. Key Results

1. Theorem 1 (Uniqueness): Uniqueness of entropy solutions is established with a stability estimate dependent on the initial data and the Lipschitz constants of the flux and nonlocal functions.
2. Theorem 3 (Convergence): The finite volume approximations converge to the unique entropy solution of the Cauchy problem.
3. Theorem 6 (Exact Solution Limit): As $\delta \to 0$, $\|U_\delta(T) - U(T)\|_{L^1} = O(\sqrt{\delta})$, where $U_\delta$ is the solution with memory and $U$ is the solution without memory.
4. Theorem 7 (Numerical Limit): The numerical solution $U^\Delta_\delta$ converges to the memoryless solution $U$ with rate $O(\sqrt{\delta} + \sqrt{\Delta x})$, provided the temporal kernel has a finite first moment.
5. Numerical Experiments:
   • Simulations on a Keyfitz-Kranzer system type model confirm the theoretical convergence rates.
   • Observed convergence rates in experiments were slightly better ($\alpha > 0.5$) than the theoretical $O(\sqrt{\delta})$ bound, validating the robustness of the scheme.
   • Visualizations demonstrate the transition from "memory" dynamics to "memoryless" dynamics as $\delta$ decreases.

5. Significance and Future Directions

Significance:

• Theoretical Gap Filled: This work bridges a critical gap in the theory of hyperbolic conservation laws by rigorously handling systems with both spatial and temporal nonlocality.
• Practical Modeling: The results are directly applicable to modeling phenomena with fading memory, such as traffic flow with driver reaction times, viscoelasticity, and gas transport in porous media.
• Numerical Reliability: By proving asymptotic compatibility, the authors ensure that numerical simulations of memory-based models will correctly approximate the simpler, memoryless models in the limit, preventing numerical artifacts when memory effects are negligible.

Future Directions & Open Problems:

• Nonlocal-in-Time Only: The case where spatial nonlocality is removed (purely temporal memory) remains open for nonlinear fluxes, as the system behaves like a local law with discontinuous flux, losing spatial BV bounds.
• Fully Local Limit: The complete passage from the nonlocal space-time system to a fully local system (removing both $\mu$ and $\Gamma$) for $N > 1$ with nonlinear fluxes remains an open and difficult problem due to the ill-posedness of the corresponding local coupled systems.
• Extensions: Future work will address source terms, less regular kernels, and error analysis for the nonlocal-to-local transition.