  We use the general exact solution of the Cauchy problem for the compressible
Euler vortex equation in unbounded space which was obtained earlier
(S.G.Chefranov, Sov. Phys. Dokl., 36, 286, 1991). This solution loses its
smoothness in finite time and coincides with the exact solution of the Hopf
equation, describing the inertial motion of the ideal fluid without pressure.
On this base we obtain here the new smooth at all times solution to the
compressible Navier-Stokes (NS) equation with the pressure field shows linear
proportionality to the divergence of the velocity field, as it is known for an
out-of-equilibrium systems with large second viscosity and small first
viscosity. For example, directly from this solution of the NS equation for the
case of two-dimensional (2D) compressible flow the exact representation of
energy spectrum well known for 2D incompressible case (R.H.Kraichnan,
Phys.Fluids,vol.10,1417,1967) is obtained.
