Gas-kinetic derivation of Navier-Stokes-like traffic equations
  Macroscopic traffic models have recently been severely criticized to base on lax analogies only and to have a number of deficiencies. Therefore, this paper shows how to construct a logically consistent fluid-dynamic traffic model from basic laws for the acceleration and interaction of vehicles. These considerations lead to the gas-kinetic traffic equation of Paveri-Fontana. Its stationary and spatially homogeneous solution implies equilibrium relations for the `fundamental diagram', the variance-density relation, and other quantities which are partly difficult to determine empirically.   Paveri-Fontana's traffic equation allows the derivation of macroscopic moment equations which build a system of non-closed equations. This system can be closed by the well proved method of Chapman and Enskog which leads to Euler-like traffic equations in zeroth-order approximation and to Navier-Stokes-like traffic equations in first-order approximation. The latter are finally corrected for the finite space requirements of vehicles. It is shown that the resulting model is able to withstand the above mentioned criticism. 
Dirk Helbing