Singular limit for a class of nonlocal conservation laws via compensated compactness
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arXiv
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| Format: | Preprint |
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2025
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| author | Coclite, Giuseppe Maria De Nitti, Nicola Huang, Kuang |
| author_facet | Coclite, Giuseppe Maria De Nitti, Nicola Huang, Kuang |
| contents | 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 Ole\uınik-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15631 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singular limit for a class of nonlocal conservation laws via compensated compactness Coclite, Giuseppe Maria De Nitti, Nicola Huang, Kuang Analysis of PDEs 35L65 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 Ole\uınik-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. |
| title | Singular limit for a class of nonlocal conservation laws via compensated compactness |
| topic | Analysis of PDEs 35L65 |
| url | https://arxiv.org/abs/2511.15631 |