Applied Physics and Mathematics Annotation << Back EXACT NOT STATIONARY SOLUTION OF THE EULER-HELMHOLTZ AND RIEMANN-HOPF THREE-DIMENSIONAL EQUATIONS FOR VORTEX FLOW OF COMPRESSIBLE MEDIUM AND THE SIXTH MILLENNIUM PROBLEM S.G. CHEFRANOV, A.S. CHEFRANOV Exact analytical solution of the Cauchy problem in unlimited space to three-dimensional Euler-Helmholtz (EG) equations when nonzero divergence of velocity fi eld had been received. It describes the rotational inertia of an ideal compressible medium and coincides with the exact solution of the threedimensional Riemann-Hopf (RH) equation simulating turbulence without pressure (S.G. Chefranov, 1991). A necessary and suffi cient condition for the occurrence of singularities in the evolution of enstrofy for the end time t = t0, whenever possible its continuance on times t ≥ t0 in Sobolev space H0(R3), but it is impossible to do this in H1(R3), had been received for this solution. Closed description of the enstrofy evolution and higher moments of the velocity and vorticity fi elds was given; the problem of circuit theory of turbulence was exactly solved. The possibility of continuing the resulting smooth solutions of equations EG and RH in the Sobolev space Hq(R3) already have for any q ≥ 1 and t ≥ t0 (due to introduction of an enough large coeffi cient of linear external friction or due to introduction of arbitrarily small effective bulk viscosity) was shown. The new divergent solution of the Cauchy problem for threedimensional Navier-Stokes (NS) equation which is the same as specifi ed smooth solution of equations EG and RH (if any q ≥ 1 and t ≥ t0), taking into account the effect of viscosity, compressible medium and a suffi cient condition for positivity of integral entropy growth rate as a linear connection of pressure and velocity fi eld divergence, was received. The irrationality of the a priori exclusion of divergent decisions in the defi nition of the sixth Millennium problems in the Clay Mathematical Institute (Cambridge, MA), when you want to solve the problem of the existence of smooth solutions of the NS three-dimensional equation only for the case of no divergent currents incompressible medium, was proven. Key words: hydrodynamics, compressibility, viscosity, turbulence, vortex waves. Contacts: E-mail: schefranov@mail.ru, E-mail: a.chef@bk.ru Pp. 18-32.