  The system of Navier--Stokes--Fourier equations is one of the most celebrated
systems of equations in modern science. It describes dynamics of fluids in the
limit when gradients of density, velocity and temperature are sufficiently
small, and loses its applicability when the flux becomes so non-equilibrium
that the changes of velocity, density or temperature on the length compatible
with the mean free path are non-negligible. The question is: how to model such
fluxes? This problem is still open. (Despite the fact that the first `final
equations of motion' modified for analysis of thermal creep in rarefied gas
were proposed by Maxwell in 1879.) There are, at least, three possible answers:
(i) use molecular dynamics with individual particles, (ii) use kinetic
equations, like Boltzmann's equation, or (iii) find a new system of equations
for description of fluid dynamics with better accounting of non-equilibrium
effects. These three approaches work at different scales. We explore the third
possibility using the recent findings of capillarity of internal layers in
ideal gases and of saturation effect in dissipation (there is a limiting
attenuation rate for very short waves in ideal gas and it cannot increase
infinitely). One candidate equation is discussed in more detail, the Korteweg
system proposed in 1901. The main ideas and approaches are illustrated by a
kinetic system for which the problem of reduction of kinetics to fluid dynamics
is analytically solvable.
