The Exact Solution to the 3D Vortex Compressible Euler Equation and the Clay Millennium Problem Generalization

Exact solution of the compressible Euler–Helmholtz equation and the millennium prize problem generalization

“…We also note that in the general case the vortical solutions to the three-dimensional RH equation coincide with the solutions to the three-dimensional EH equation for describing vortex flows of an ideal compressible medium with nonzero divergence of the velocity field [9,[12][13][14][15][16] (see also appendices A and B).…”

“…In the two-dimensional case there is an exact correspondence between the criterion (4.7) and a similar criterion given in [27] (see formula (9) in [27]) in connection with the solution of the problem of propagation a flame front (generated by a self-sustained exothermal chemical reaction) on the basis of the simplified version of the Sivashisky equation [28]:…”

“…2. The question of existence of smooth time-dependent nonzero-divergence and divergence-free solutions to the three-dimensional NS equation in unbounded space (or in space with periodic boundary conditions) and on an unbounded time interval remains open during almost two hundred years (starting from 1827-1845) [3,[7][8][9][10][11].…”

“…The importance of this problem is determined by not only purely mathematical but also practical interest in connection with the basic and applied problem of predictability of hydro-meteorological and other fields developed in using the methods of numerical solution of the NS equation [8,9]. In this connection, in 2000 the problem of existence of a smooth time-dependent vortical solution with finite initial energy to the three-dimensional NS equation on an unbounded time interval was stated by the Clay Mathematics Institute as one of seven basic Millennium Prize Problems [7].…”

“…In fact, a new analytic time-dependent vortex solution to the complete 3D NS equation is obtained here on the basis of exact vortex solution for the 3D Riemann-Hopf (RH) equation and its modifications which takes into account the effective viscosity or friction [12][13][14][15][16] (see also appendix A).…”