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

Abstract: We consider a class of nonlocal conservation laws modeling traffic flows, given by $$\partial_t u_\varepsilon + \partial_x(V(u_\varepsilon \ast γ<em>\varepsilon) u</em>\varepsilon) = 0$$, with a rescaled convolution kernel $$γ<em>\varepsilon(\cdot) := \varepsilon<sup>{-1}γ(\cdot/\varepsilon)$$. We establish the strong $\mathrm L¹</em>{\mathrm{loc}}$-convergence of weak solutions $u_\varepsilon$ toward the entropy-admissible solution of the corresponding local conservation law as the kernel $γ<em>\varepsilon$ concentrates to a Dirac delta distribution when $\varepsilon \searrow 0$. In contrast to previous literature, we obtain compactness of the family ${u</em>\varepsilon \ast γ<em>\varepsilon}</em>{\varepsilon&gt;0}$ without relying on total variation bounds or Oleĭnik-type estimates. Instead, we establish $\mathrm L<sup>2$-type bounds on its entropy production and use the theory of compensated compactness, assuming that the initial datum merely belongs to $\mathrm L^1\cap \mathrm L<sup>\infty$. 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.