The Navier Stokes system models the dynamics of a viscous incompressible uid. The problem of existence of solutions of the Cauchy Dirichlet problem for this system is included in the list of the most serious problems of this century. In this paper it is proposed to consider the multipoint initial-nal conditions instead of the Cauchy conditions. It should be noted that nowadays the study of solvabilityof initial-nal value problems is a new and actively developing direction of the Sobolev type equations theory. The main result of the paper is the proof of unique solvability of the stated problem for the system of Navier Stokes equations.

The problems of distribution of transport flows are currently relevant in connection with the increase in vehicles. In the 50s of the last century, the first macroscopic (hydrodynamic) models appeared, where the transport flow resembles the flow "motivated" compressible liquid. The scientific approach based on the Navier-Stokes system. The main idea of the scholars is consideration the hydrodynamic models on the grounds of interrelation between the transport flow and incompressible fluid. For modelling traffic flows we examine Oskolkov equation on the geometric graph, where the edge has two positive values corresponding to it "length" and "width". Certainly, in the context of mathematical model the values l k and b k are dimensionless, but for clarity it is convenient to imagine that l k is measured in linear metric units, for example, kilometers or miles, and b k is equal to the number of traffic lanes on the roadway in one direction. In terms of the Oskolkov model, we obtained a nonclassical multipoint initial-final value condition. We will study such a model using the idea and methods of the Sobolean equation theory. These notes describe a numerical experiment based on the Galerkin method for the Oskolkov equation with a multipoint initial-final condition on the graph.

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