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For the first time the exact vortex solution of the Cauchy problem inunbounded space is obtained for the three-dimensional Euler-Helmholtz (EH)equation in the case of a nonzero-divergence velocity field for an idealcompressible medium. The solution obtained describes the inertial vortex motionand coincides with the exact solution to the three-dimensional Riemann-Hopf(RH) equation which simulates turbulence without pressure [Chefranov, 1991]. Anecessary and sufficient condition of the onset of a singularity in theevolution of the enstrophy in finite time t=t0 is obtained for this solutionwhen its continuation is possible in times t>=t0 in the Sobolev space H^0(R^3)but cannot be made in H^1(R^3). A closed description of the evolution of theenstrophy and the all other moments of the velocity and vortex fields is given,i.e., the main problem of theory of turbulence is solved exactly. Thepossibility of continuation of the obtained smooth solution to the EH an RHequations in the Sobolev space H^q(R^3) is also demonstrated for any q>=1 andt>=t0 due to introduction of a fairly large homogeneous friction or byintroducing an arbitrary small effective volume viscosity. A new analyticsolution of the Cauchy problem for the three-dimensional Navier-Stokes (NS)equation is obtained. This solution coincides with the above-mentioned smoothsolution to the EH and RH equations, which take into account the viscosityeffect of a compressible medium and also the sufficient condition of positivedefiniteness of the growth rate of the entropy in the form of a linear relationbetween the pressure and the divergence of the velocity field. This gives thepositive solution to the generalization of the Millennium Prize Problem on thecompressible Navier-Stokes equation where only the modification of viscousforce is used (by introduction of homogeneous friction and etc.).

physica scripta

Exact Time-Dependent Solution to the Three-Dimensional Euler- Helmholtz  and Riemann-Hopf Equations for Vortex Flow of a Compressible Medium and the  Sixth Millennium Prize Problem