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Nonlocal transport equations – existence and uniqueness of solutions and relation to the corresponding conservation laws. (English) Zbl 1471.35250

The authors study the existence and uniqueness of solutions to 1D nonlocal transport equations, where the nonlocal term consists of an integration of the flux function’s derivative over a neighborhood of the corresponding space-time coordinate. Using a fixed-point argument in the characteristics, the authors establish the existence and uniqueness of weak solutions for initial datum in \(L^\infty ({\mathbb R}) \cap TV ({\mathbb R})\). Moreover, it is shown that as the nonlocal reach tends to zero, the nonlocal transport equation converges to the corresponding local conservation law in the case where the Dirac distribution is approached in a symmetric way and the flux function is quadratic (including Burgers’ equation and the Lighthill-Whitham-Richards traffic flow model). And for specific quasi-convex and quasi-concave initial datum, even the convergence to the local entropy solution is obtained. On the other hand, some counter examples are given which show that for “nonsymmetric” nonlocal approximations, the solution cannot converge to the entropy solution or even a weak solution. Finally, some numerical examples are presented which demonstrate that this convergence appears to hold even for more general initial datum.

MSC:

35Q49 Transport equations
35L03 Initial value problems for first-order hyperbolic equations
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
76L05 Shock waves and blast waves in fluid mechanics
35D30 Weak solutions to PDEs
35B35 Stability in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
90B20 Traffic problems in operations research
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[1] M. A\u\izerman, E. Bredihina, S. Černikov, F. Gantmaher, I. Gelfand, S. Gelfer, D. Harazov, M. Kadec, J. Korobe\u\inik, M. Kre\u\in, O. Ole\u\inik, I. Pyatecki\u\i-Šapiro, M. Subhankulov, K. Temko, and A. Turecki\u\i, Seventeen Papers on Analysis, American Mathematical Society, Providence, RI, 1963.
[2] P. Amorim, R. Colombo, and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM Math. Model. Numer. Anal., 49 (2015), pp. 19-37. · Zbl 1317.65165
[3] D. Armbruster, D. Marthaler, C. Ringhofer, K. Kempf, and T.-C. Jo, A continuum model for a re-entrant factory, Oper. Res., 54 (2006), pp. 933-950, http://dblp.uni-trier.de/db/journals/ior/ior54.html#ArmbrusterMRKJ06. · Zbl 1167.90477
[4] G. Baker, X. Li, and A. Morlet, Analytic structure of two \textup1d-transport equations with nonlocal fluxes, Phys. D, 91 (1996), pp. 349-375, https://doi.org/10.1016/0167-2789(95)00271-5. · Zbl 0899.76104
[5] P. Balodis and A. Córdoba, An inequality for riesz transforms implying blow-up for some nonlinear and nonlocal transport equations, Adv. Math., 214 (2007), pp. 1-39, https://doi.org/10.1016/j.aim.2006.07.021. · Zbl 1133.35078
[6] S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), pp. 217-241. · Zbl 1336.65130
[7] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford University Press, Oxford, 2000. · Zbl 0977.35087
[8] A. Bressan and W. Shen, On Traffic Flow with Nonlocal Flux: A Relaxation Representation, preprint, https://arxiv.org/abs/1911.03636, 2019. · Zbl 1446.35072
[9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. · Zbl 1220.46002
[10] F. Chiarello and P. Goatin, Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel, ESAIM Math. Model. Numer. Anal., 52 (2018), pp. 163-180. · Zbl 1395.35142
[11] M. Colombo, G. Crippa, and L. Spinolo, On the singular local limit for conservation laws with nonlocal fluxes, Arch. Ration. Mech. Anal., 233 (2019), pp. 1131-1167. · Zbl 1415.35188
[12] M. Colombo, G. Crippa, and L. Spinolo, Local Limit of Nonlocal Traffic Models: Convergence Results and Total Variation Blow-up, preprint, https://arxiv.org/abs/1808.03529, 2019.
[13] R. Colombo, M. Garavello, and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023. · Zbl 1248.35213
[14] A. Córdoba, D. Córdoba, and M. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2), 162 (2005), pp. 1377-1389. · Zbl 1101.35052
[15] J.-M. Coron, M. Kawski, and Z. Wang, Analysis of a conservation law modeling a highly re-entrant manufacturing system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), pp. 1337-1359, https://doi.org/10.3934/dcdsb.2010.14.1337. · Zbl 1207.35218
[16] J.-M. Coron and Z. Wang, Controllability for a scalar conservation law with nonlocal velocity, J. Differential Equations, 252 (2012), pp. 181-201. · Zbl 1243.35165
[17] C. De Lellis, P. Gwiazda, and A. Świerczewska-Gwiazda, Transport equations with integral terms: existence, uniqueness and stability, Calc. Var. Partial Differential Equations, 55 (2016), 128. · Zbl 1404.82094
[18] D. del Castillo-Negrete, Fractional diffusion models of nonlocal transport, Phys. Plasmas, 13 (2006), 082308, https://doi.org/10.1063/1.2336114.
[19] M. D. Francesco, S. Fagioli, and E. Radici, Deterministic particle approximation for nonlocal transport equations with nonlinear mobility, J. Differential Equations, 266 (2019), pp. 2830-2868. · Zbl 1428.35608
[20] M. Gugat, A. Keimer, G. Leugering, and Z. Wang, Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks, Netw. Heterog. Media, 10 (2015), pp. 749-785, https://doi.org/10.3934/nhm.2015.10.749. · Zbl 1335.49011
[21] M. Haderlein, D. Segets, M. Gröschel, L. Pflug, G. Leugering, and W. Peukert, FIMOR: An efficient simulation for ZnO quantum dot ripening applied to the optimization of nanoparticle synthesis, Chem. Eng. J., 260 (2015), pp. 706-715.
[22] J. Hale, Ordinary Differential Equations, R. E. Krieger Pub., Huntington, NY, 1980. · Zbl 0433.34003
[23] J. Jahn, Introduction to the Theory of Nonlinear Optimization, 3rd ed., Springer, Berlin, 2007. · Zbl 1115.49001
[24] A. Keimer and L. Pflug, Existence, uniqueness and regularity results on nonlocal balance laws, J. Differential Equations, 263 (2017), pp. 4023-4069. · Zbl 1372.35186
[25] A. Keimer and L. Pflug, On approximation of local conservation laws by nonlocal conservation laws, J. Math. Anal. Appl., 475 (2019), pp. 1927-1955. · Zbl 1428.35213
[26] A. Keimer, L. Pflug, and M. Spinola, Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping, J. Math. Anal. Appl., 466 (2018), pp. 18-55, https://doi.org/10.1016/j.jmaa.2018.05.013. · Zbl 1394.35275
[27] A. Keimer, L. Pflug, and M. Spinola, Nonlocal balance laws: Theory of convergence for nondissipative numerical schemes, submitted. · Zbl 1394.35275
[28] A. Keimer, L. Pflug, and M. Spinola, Nonlocal scalar conservation laws on bounded domains and applications in traffic flow, SIAM J. Math. Anal., 50 (2018), pp. 6271-6306, https://doi.org/10.1137/18M119817X. · Zbl 1404.35274
[29] S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, 2nd ed., Birkhäuser Boston, Boston, 2002. · Zbl 1015.26030
[30] G. Leoni, A First Course in Sobolev Spaces, Grad. Stud. Math. 105, American Mathematical Society, Providence, RI, 2009. · Zbl 1180.46001
[31] B. Piccoli, N. Duteil, and E. Trélat, Sparse control of Hegselmann-Krause models: Black hole and declustering, SIAM J. Control Optim., 57 (2019), pp. 2628-2659, https://doi.org/10.1137/18M1168911. · Zbl 1422.91620
[32] B. Piccoli and F. Rossi, Transport equation with nonlocal velocity in Wasserstein spaces: Convergence of numerical schemes, Acta Appl. Math., 124 (2013), pp. 73-105. · Zbl 1263.35202
[33] B. Piccoli, F. Rossi, and M. Tournus, A Wasserstein Norm for Signed Measures, with Application to Nonlocal Transport Equation with Source Term, preprint, https://arxiv.org/abs/1910.05105, 2019.
[34] V. Skorych, M. Dosta, E.-U. Hartge, and S. Heinrich, Novel system for dynamic flowsheet simulation of solids processes, Powder Technology, 314 (2017), pp. 665-679.
[35] B. Uhrin, Some remarks about the convolution of unimodal functions, Ann. Probab., 12 (1984), pp. 640-645. · Zbl 0554.60028
[36] W. Walter, Differential and Integral Inequalities, Springer-Verlag, Berlin, New York, 1970. · Zbl 0252.35005
[37] E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, Appl. Math. Sci. 108, Springer-Verlag, New York, 1995. · Zbl 0834.46002
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