Nonlocal multi-scale traffic flow models: analysis beyond vector spaces

Abstract

Realistic models of traffic flow are nonlinear and involve nonlocal effects in balance laws. Flow characteristics of different types of vehicles, such as cars and trucks, need to be described differently. Two alternatives are used here, L p -valued Lebesgue measurable density functions and signed Radon measures. The resulting solution spaces are metric spaces that do not have a linear structure, so the usual convenient methods of functional analysis are no longer applicable. Instead ideas from mutational analysis will be used, in particular the method of Euler compactness will be applied to establish the well-posedness of the nonlocal balance laws. This involves the concatenation of solutions of piecewise linear systems on successive time subintervals obtained by freezing the nonlinear nonlocal coefficients to their values at the start of each subinterval. Various compactness criteria lead to a convergent subsequence. Careful estimates of the linear systems are needed to implement this program.

Nonlocal hyperbolic population models structured by size and spatial position: Well-posedness

“…Finally, we regard this initial value problem as a representative of a quite broad problem class in which nonlocal nonlinear hyperbolic equations meet a boundary condition concerning one real component at only one boundary point of its interval. Hence, our approach to well-posedness is based on combining and extending preceding achievements in [64] and [70, § 2.6]. Concluding existence of solutions from Euler's method in combination with sequential compactness has the significant advantage that the results can be extended to systems rather easily (see, e.g., [70] for more details).…”

“…The parameter p ∈ (1, ∞) decides which class of singularities is still covered. This notion has already proved useful for analytical frameworks modelling cancer cell migration [74] and traffic flow [64].…”

“…Hence, we aim at a problem class in which nonlocal effects can be taken into consideration immediatelynot necessarily restricted to convolution operators. In various contexts of modelling, nonlocal differential problems of hyperbolic type have already been investigated thoroughly by Colombo, Goatin and collaborators, for example (see [2,15,21,22,23,20,25,44,64,74] and references therein). Results about measure-valued solutions, in particular, are usually based on such functional dependence of coefficients (see, e.g., [17,18,24,47,64,70,78,79,84] and references therein).…”

“…a metric D on L p ([s min , s max ]×R ) which metrizes the weak convergence on norm bounded tight subsets. (Similar approaches are used in, e.g., [47,64,70,74]. )…”

“…The main result about existence is formulated in subsequent Theorem 2.2. Similarly to Peano's existence theorem for ordinary differential equations, it is proved by a modified Euler method using solutions to piecewise autonomous linear problems in combination with a special form of sequential compactness (called Euler compactness in subsection 5.8 below, see also, e.g., [47,62,64,70,71,73]). The advantage of this way to existence consists in the quite weak regularity assumptions for the coefficients (in a word, they are just supposed to be appropriately bounded and continuous).…”