# American Institute of Mathematical Sciences

• Previous Article
On the well-posedness of the inviscid multi-layer quasi-geostrophic equations
• DCDS Home
• This Issue
• Next Article
Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $h\mathbb{Z}$
June  2019, 39(6): 3197-3213. doi: 10.3934/dcds.2019132

## Regularity results for the solutions of a non-local model of traffic flow

 1 Laboratoire J. A. Dieudonné, UMR 7351 CNRS, Université Côte d'Azur, LJAD, CNRS, Inria, France 2 Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice cedex 2, France 3 Inria Sophia Antipolis - Méditerranée, Université Côte d'Azur, Inria, CNRS, LJAD, 2004 route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France

* Corresponding author: Florent Berthelin

Received  June 2018 Revised  November 2018 Published  February 2019

Fund Project: The first author was partially supported by Inria

We consider a non-local traffic model involving a convolution product. Unlike other studies, the considered kernel is discontinuous on $\mathbb R$. We prove Sobolev estimates and prove the convergence of approximate solutions solving a viscous and regularized non-local equation. It leads to weak, ${{\bf{C^{}}}}([0,T], {{\bf{L^2}}}( \mathbb R))$, and smooth, ${\bf{W}}^{2,2N}([0,T]\times \mathbb R)$, solutions for the non-local traffic model.

Citation: Florent Berthelin, Paola Goatin. Regularity results for the solutions of a non-local model of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3197-3213. doi: 10.3934/dcds.2019132
##### References:

show all references

##### References:
 [1] Jan Friedrich, Oliver Kolb, Simone Göttlich. A Godunov type scheme for a class of LWR traffic flow models with non-local flux. Networks & Heterogeneous Media, 2018, 13 (4) : 531-547. doi: 10.3934/nhm.2018024 [2] Felisia Angela Chiarello, Paola Goatin. Non-local multi-class traffic flow models. Networks & Heterogeneous Media, 2019, 14 (2) : 371-387. doi: 10.3934/nhm.2019015 [3] Maria Laura Delle Monache, Paola Goatin. Stability estimates for scalar conservation laws with moving flux constraints. Networks & Heterogeneous Media, 2017, 12 (2) : 245-258. doi: 10.3934/nhm.2017010 [4] Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617 [5] Adimurthi , Shyam Sundar Ghoshal, G. D. Veerappa Gowda. Exact controllability of scalar conservation laws with strict convex flux. Mathematical Control & Related Fields, 2014, 4 (4) : 401-449. doi: 10.3934/mcrf.2014.4.401 [6] Darko Mitrovic. New entropy conditions for scalar conservation laws with discontinuous flux. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1191-1210. doi: 10.3934/dcds.2011.30.1191 [7] Paola Goatin, Sheila Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Networks & Heterogeneous Media, 2016, 11 (1) : 107-121. doi: 10.3934/nhm.2016.11.107 [8] Shijin Deng, Weike Wang. Pointwise estimates of solutions for the multi-dimensional scalar conservation laws with relaxation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1107-1138. doi: 10.3934/dcds.2011.30.1107 [9] Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159 [10] Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255 [11] Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73 [12] Anupam Sen, T. Raja Sekhar. Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045 [13] Julien Jimenez. Scalar conservation law with discontinuous flux in a bounded domain. Conference Publications, 2007, 2007 (Special) : 520-530. doi: 10.3934/proc.2007.2007.520 [14] Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2319-2347. doi: 10.3934/dcds.2013.33.2319 [15] Laurent Lévi, Julien Jimenez. Coupling of scalar conservation laws in stratified porous media. Conference Publications, 2007, 2007 (Special) : 644-654. doi: 10.3934/proc.2007.2007.644 [16] Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 751-759. doi: 10.3934/nhm.2007.2.751 [17] Darko Mitrovic. Existence and stability of a multidimensional scalar conservation law with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (1) : 163-188. doi: 10.3934/nhm.2010.5.163 [18] John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1501-1526. doi: 10.3934/cpaa.2013.12.1501 [19] Sanda Cleja-Ţigoiu, Raisa Paşcan. Non-local elasto-viscoplastic models with dislocations and non-Schmid effect. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1621-1639. doi: 10.3934/dcdss.2013.6.1621 [20] Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello. Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5913-5942. doi: 10.3934/dcds.2017257

2018 Impact Factor: 1.143