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#### Keywords

- fixpoint theorem (1)
- kinetic models (1)
- stationary solutions (1)
- vehicular traffic (1)

We discuss how kinetic and aerodynamic descriptions of a gas can be matched at some prescribed boundary. The boundary (matching) conditions arise from requirement that the relevant moments (p,u,...) of the particle density function be continuous at the boundary, and from the requirement that the closure relation, by which the aerodynamic equations (holding on one side of the boundary) arise from the kinetic equation (holding on the other side), be satisfied at the boundary. We do a case study involving the Knudsen gas equation on one side and a system involving the Burgers equation on the other side in section 2, and a discussion for the coupling of the full Boltzmann equation with the compressible Navier-Stokes equations in section 3.

In this paper the kinetic model for vehicular traffic developed in [3,4] is considered and theoretical results for the space homogeneous kinetic equation are presented. Existence and uniqueness results for the time dependent equation are stated. An investigation of the stationary equation leads to a boundary value problem for an ordinary differential equation. Existence of the solution and some properties are proved. A numerical investigation of the stationary equation is included.