}, author = {Coclite, Giuseppe Maria and De Nitti, Nicola and Keimer, Alexander and Pflug, Lukas and Zuazua Iriondo, Enrique}, doi = {10.1088/1361-6544/acf01d}, faupublication = {yes}, journal = {Nonlinearity}, keywords = {Nonlocal conservation laws; nonlocal flux; Burgers’ equation; approximation of local conservation laws; N-waves; source-type solutions; entropy solutions}, peerreviewed = {Yes}, title = {{Long}-time convergence of a nonlocal {Burgers}' equation towards the local {N}-wave}, url = {https://iopscience.iop.org/article/10.1088/1361-6544/acf01d}, year = {2023} } @incollection{faucris.288797419, abstract = {In this contribution we give a short overview of recent advances in nonlocal conservation laws. In these equations, we assume that the flux f is decomposable into f(x)=V(x)x, x∈R with V being the velocity function. The velocity itself depends nonlocally on the solution, and nonlocality here is realized by an integration over a specific spatial neighborhood. We tackle first problems of existence, uniqueness and regularity of solutions, dwell on the fact that weak solutions are unique (and an entropy condition is not required), and discuss delay nonlocal conservation laws as well as multi-d equations. One key question of nonlocal conservation laws has been whether solutions to these equations converge to the solution of the corresponding local conservation laws when the nonlocality is shrunk to a local evaluation. This has recently been answered positively for a variety of nonlocal impacts, thus closing the gap between nonlocal and local modeling with conservation laws. In contrast to the character of viscosity approximations being parabolic, we keep with the nonlocal approximation a hyperbolic character of the dynamics. This opens a different and novel approach for tackling nonlocal conservation laws.}, author = {Keimer, Alexander and Pflug, Lukas}, doi = {10.1016/bs.hna.2022.11.001}, faupublication = {yes}, keywords = {Discontinuous nonlocal conservation laws; Fixed-point methods; Nonlocal conservation laws; Population balance equations; Singular limit problem; Traffic flow modeling}, note = {CRIS-Team Scopus Importer:2023-02-03}, peerreviewed = {unknown}, publisher = {Elsevier B.V.}, series = {Handbook of Numerical Analysis}, title = {{Nonlocal} balance laws – an overview over recent results}, year = {2023} } @article{faucris.216448317, abstract = {We consider a nonlocal conservation law on a bounded spatial domain and show existence and uniqueness of weak solutions for nonnegative flux function and left-hand-side boundary datum. The nonlocal term is located in the flux function of the conservation law, averaging the solution by means of an integral at every spatial coordinate and every time, forward in space. This necessitates the prescription of a kind of right-hand-side boundary datum, the external impact on the outflow. The uniqueness of the weak solution follows without prescribing an entropy condition. Allowing the velocity to become zero (also dependent on the nonlocal impact) offers more realistic modeling and significantly higher applicability. The model can thus be applied to traffic flow, as suggested for unbounded domains in [S. Blandin and P. Goatin, Numer. Math., 132 (2016), pp. 217-241], [P. Goatin and S. Scialanga, Netw. Hetereog. Media, 11 (2016), pp. 107-121]. It possesses finite acceleration and can be interpreted as a nonlocal approximation of the famous "local" Lighthill-Whitham-Richards model [M. Lighthill and G. Whitham, Proc. Roy. Soc. London Ser. A, 229 (1955), pp. 281-316], [P. I. Richards, Oper. Res., 4 (1956), pp. 42-51]. Several numerical examples are presented and discussed also with respect to the reasonableness of the required assumptions and the model itself.}, author = {Pflug, Lukas and Spinola, Michele and Keimer, Alexander}, doi = {10.1137/18M119817X}, faupublication = {yes}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {first order macroscopic traffic flow model with finite acceleration;nonlocal conservation law;initial boundary value problem;method of characteristics;fixed-point problem}, month = {Jan}, pages = {6271-6306}, peerreviewed = {Yes}, title = {{NONLOCAL} {SCALAR} {CONSERVATION} {LAWS} {ON} {BOUNDED} {DOMAINS} {AND} {APPLICATIONS} {IN} {TRAFFIC} {FLOW}}, volume = {50}, year = {2018} } @article{faucris.216448800, author = {Pflug, Lukas and Keimer, Alexander}, doi = {10.1016/j.jmaa.2019.03.063}, faupublication = {yes}, journal = {Journal of Mathematical Analysis and Applications}, keywords = {Convergence of the nonlocal model to corresponding local model; LWR PDE; Entropy solution; Traffic flow modelling; Nonlocal balance laws}, pages = {1927-1955}, peerreviewed = {Yes}, title = {{On} approximation of local conservation laws by nonlocal conservation laws}, volume = {475}, year = {2019} } @article{faucris.314051115, abstract = {In this contribution, we study the singular limit problem of a nonlocal conservation law with a discontinuity in space. The corresponding local equation can be transformed diffeomorphically to a classical scalar conservation law to which the well-known Kružkov theory can be applied. However, the nonlocal equation does not scale that way, which is why the study of convergence is interesting to pursue. For exponential kernels in the nonlocal operator, we establish convergence to the solution of the corresponding local equation under mild conditions on the discontinuous velocity. We illustrate our results with some numerical examples.}, author = {Keimer, Alexander and Pflug, Lukas}, doi = {10.1016/j.na.2023.113381}, faupublication = {yes}, journal = {Nonlinear Analysis - Theory Methods & Applications}, keywords = {Convergence nonlocal to local; Discontinuous nonlocal conservation law; Entropy solution; Existence and uniqueness; Nonlocal conservation law; Singular limit}, note = {CRIS-Team Scopus Importer:2023-11-17}, peerreviewed = {Yes}, title = {{On} the singular limit problem for a discontinuous nonlocal conservation law}, volume = {237}, year = {2023} } @article{faucris.112053744, abstract = {The Lavrentiev regularization method is a tool to improve the regularity of the Lagrange multipliers in pde constrained optimal control problems with state constraints. It has already been used for problems with parabolic and elliptic systems. In this paper we consider Lavrentiev regularization for problems with a hyperbolic system, namely the scalar wave equation. We show that also in this case the regularization yields multipliers in the Hilbert space L