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

Abstract

Summary For 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.

Maximum entropy production as a necessary admissibility condition for the fluid Navier–Stokes and Euler equations

“…Axisymmetric flow in 3D is also two-dimensional. Recently, an exact solution to the compressible 3D Navier-Stokes equations was obtained by using a relation between the pressure and the divergence of the velocity, which was in turn derived from the positive-definiteness of the entropy production rate, a weaker form of MEPP [9,10].…”

Exact solution to the main turbulence problem for a compressible medium and the universal −8/3 law turbulence spectrum of breaking waves

“…From (1.5) and (1.7), in particular, adding and subtracting one equation from another gives the Hopf equation [2][3][4][5][6]:…”

“…1. Let us get from (1.10) an explicit form for the velocity field using the approach developed in [2][3][4][5][6].…”

“…In our paper, a solution to all this problems is given based on the exact solution for 1-D compressible Euler equations in the explicit form, which is more useful than Riemann's solution. By using the generalized functions approach developed in [4][5][6], we obtain an explicit analytical solution of the 1-D compressible Euler equations, which gives an explicit analytical description of a nonlinear simple wave and its time of collapse for arbitrary smooth initial conditions. It is important to note that now the role of a pressure or the local speed of sound is accurately taken into account, in contrast to considerations which are provided in the frame of the Burgers equation in [1], [18], [19] and in the frame of the Hopf equation [2][3][4][5][6].…”

Some Exact Solutions of Compressible and Incompressible Euler Equations

“…The same solution in our work, is given. Moreover, people also consider other solution with befitting conditions, for example [7,8].…”