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

Sergey G. Chefranov; Artem S. Chefranov

doi:10.1088/1402-4896/aaf918

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arXiv · 1703.07239

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

Sergey G. Chefranov·Artem S. Chefranov

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Abstract

For the first time the exact vortex solution of the Cauchy problem in unbounded space is obtained for the three-dimensional Euler-Helmholtz (EH) equation in the case of a nonzero-divergence velocity field for an ideal compressible medium. The solution obtained describes the inertial vortex motion and coincides with the exact solution to the three-dimensional Riemann-Hopf (RH) equation which simulates turbulence without pressure [Chefranov, 1991]. A necessary and sufficient condition of the onset of a singularity in the evolution of the enstrophy in finite time t=t0 is obtained for this solution when 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 the enstrophy 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. The possibility of continuation of the obtained smooth solution to the EH an RH equations in the Sobolev space H^q(R^3) is also demonstrated for any q>=1 and t>=t0 due to introduction of a fairly large homogeneous friction or by introducing an arbitrary small effective volume viscosity. A new analytic solution of the Cauchy problem for the three-dimensional Navier-Stokes (NS) equation is obtained. This solution coincides with the above-mentioned smooth solution to the EH and RH equations, which take into account the viscosity effect of a compressible medium and also the sufficient condition of positive definiteness of the growth rate of the entropy in the form of a linear relation between the pressure and the divergence of the velocity field. This gives the positive solution to the generalization of the Millennium Prize Problem on the compressible Navier-Stokes equation where only the modification of viscous force is used (by introduction of homogeneous friction and etc.).

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Sergey G. Chefranov, Artem S. Chefranov. 2019-12-09. 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. https://doi.org/10.1088/1402-4896%2Faaf918

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