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

Abstract

SummaryFor the first time the exact strong solution of the three-dimensional (3D) compressible Euler–Helmholtz (EH) vortex equation in unbounded space is obtained. It corresponds to the general solution obtained in Euler variables for the 3D Riemann–Hopf (RH) equation and has a singularity in finite time. The exact closed description for the evolution of all one and multi point moments of the hydrodynamic fields and their spectra is now possible and thus, the mysterious turbulence problem has the exact solution for this case. As an example, for the first time the two-dimensional (2D) energy spectrum in the inertial scale interval is obtained directly from the exact solution of compressible 2D Navier–Stokes (NS) equation. For this purpose, a smooth continuation at all times for these exact solutions to the EH and RH equations is obtained by introducing a super-threshold homogeneous friction or any even extremely low effective viscosity. A new, smooth for all time, analytic solutions to the 1D, 2D and 3D NS equations is obtained. This solution coincides with the above-mentioned modified by viscosity solutions to the EH and RH equations, when also taking into account a linear relation between the pressure and the velocity field divergence, which is well known for the systems far from the equilibrium with relatively large second viscosity. This gives the positive answer to the generalization of the millennium prize problem (www.claymath.org) for the case of the compressible NS equation. Indeed, earlier only a negative answer to the problem of the existence of smooth solutions for any finite time has been a priori considered for the compressible case with the finite initial energy of the smooth velocity field in the unbounded space.