Singular limit for a class of nonlocal conservation laws via compensated compactness

TL;DR

This work analyzes a nonlocal conservation law for traffic flow, proving that as the nonlocal horizon $\varepsilon\to0$, the nonlocal model converges strongly in $L^1_{\text{loc}}$ to the entropy solution of the corresponding local conservation law. The authors introduce an $L^2$-type entropy-production framework and apply compensated compactness to obtain strong convergence without relying on total variation bounds, assuming only $u_0\in L^1\cap L^\infty$. They establish nonlocal-to-local limits for a piecewise-constant kernel with Greenshields' velocity and for strictly monotone kernels with decreasing velocity, thereby resolving a long-standing open problem for non-convex kernels; in the exponential-kernel case they also obtain convergence of $u_\varepsilon$. The results have implications for the mathematical justification of nonlocal traffic models and provide a pathway toward analysis of asymptotically compatible numerical schemes.

Abstract

We consider a class of nonlocal conservation laws modeling traffic flows, given by $ \partial_t u_\varepsilon + \partial_x(V(u_\varepsilon \ast γ_\varepsilon) u_\varepsilon) = 0$, with a rescaled convolution kernel $γ_\varepsilon(\cdot) := \varepsilon^{-1}γ(\cdot/\varepsilon)$. We establish the strong $\mathrm L^1_{\mathrm{loc}}$-convergence of weak solutions $u_\varepsilon$ toward the entropy-admissible solution of the corresponding local conservation law as the kernel $γ_\varepsilon$ concentrates to a Dirac delta distribution when $\varepsilon \searrow 0$. In contrast to previous literature, we obtain compactness of the family $\{u_\varepsilon \ast γ_\varepsilon\}_{\varepsilon>0}$ without relying on total variation bounds or Oleĭnik-type estimates. Instead, we establish $\mathrm L^2$-type bounds on its entropy production and use the theory of compensated compactness, assuming that the initial datum merely belongs to $\mathrm L^1\cap \mathrm L^\infty$. Our results are twofold. First, we establish the nonlocal-to-local limit for the piecewise constant kernel $γ(\cdot) := {1}_{[-1,0]}(\cdot)$ combined with the affine velocity function from Greenshields' traffic model. Second, we prove the limit for strictly monotone kernels along with decreasing velocity functions. These results settle a long-standing open problem concerning the nonlocal-to-local convergence for non-convex kernels.

Theorems & Definitions (16)

• Theorem 1.1: Compactness of the entropy production
• Theorem 1.2: Convergence
• Proposition 2.1: Compensated compactness
• Lemma 2.2: Murat's compact embedding
• Proposition 3.1: Entropy balance for $w_\varepsilon$
• proof
• Remark 3.2: Entropy balance for $w_\varepsilon$ (piecewise constant kernel)
• Proposition 3.3: Entropy balance for $u_\varepsilon$ (exponential kernel)
• proof
• Lemma 4.1: Hardy--Littlewood's inequality