The Oskolkov equations on the geometric graphs as a mathematical model of the traffic flow

Currently there arose a necessity of creation of adequate mathematical model describing the flow of traffic. The mathematical traffic control theory is now actively developing in the works of A.B. Kurzhanski and his school, where the transport flow is considered to be similar to the flow of an incompressible fluid, and consequently the hydrodynamic model, for example based on the system of Navier – Stokes Equations, is used. In addition to the obvious properties of traffic flow covered previously, such as viscosity and incompressibility, the authors of this article propose to take into consideration its elasticity. Indeed, when you turn on a forbidding signal of a traffic light vehicles do not stop instantly and smoothly reduce their speed up to stop accumulating before the stop line. Similarly, if you turn on an allowing signal of the traffic light vehicles do not start instantaneously and simultaneously, they start driving one after another, gradually raising up the speed. Thus the transport flow has an effect of retardation, which is typical for viscoelastic incompressible fluids described by a system of Oskolkov equations.
The first part of the article substantiates a linear mathematical model, i.e. the model without convective terms in the Oskolkov equations. In the context of the model this means that transposition of vehicles can be neglected. In the second part the model is investigated on a qualitative level, i.e. we formulate the existence of a unique solution theorem for the stated problem and provide an outline of its proof.