NONLOCAL SCALAR CONSERVATION LAWS ON BOUNDED DOMAINS AND APPLICATIONS IN TRAFFIC FLOW

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.