  The Chapman-Enskog method is generalized for accounting the effect of kinetic
modes on hydrodynamic evolution. Hydrodynamic states of phonon system of
insulators have been studied in a small drift velocity approximation. For
simplicity, the investigation was carried out for crystals of the cubic class
symmetry. It has been found that in phonon hydrodynamics, local equilibrium is
violated even in the approximation linear in velocity. This is due to the
absence of phonon momentum conservation law that leads to a drift velocity
relaxation. Phonon hydrodynamic equations which take dissipative processes into
account have been obtained. The results were compared with the standard theory
based on the local equilibrium validity. Integral equations have been obtained
for calculating the objects of the theory (including viscosity and heat
conductivity). It has been shown that in low temperature limit, these equations
are solvable by iterations. Steady states of the system have been considered
and an expression for steady state heat conductivity has been obtained. It
coincides with the famous result by Akhiezer in the leading low temperature
approximation. It has been established that temperature distribution in the
steady state of insulator satisfies a condition of heat source absence.
