Cosmological and Quantum Solutions of the Navier–Stokes Equations

It is shown that the vector Navier–Stokes equation has a variety of quantum solutions, so the scope of this equation is not limited to the field of classical Newtonian physics, but also includes quantum physics. On this basis, it is shown that the homogeneous quantum velocity is globally defined at all moments in time, is a globally smooth and bounded function, which falls exponentially, and that the kinetic energy is also globally bounded. Its quantum solutions do not depend on the Planck constant, which is instead automatically replaced in the Navier–Stokes equation by its hydrodynamic analog $\tilde{h}=2 mD>>\mathrm{\hslash}.$ The Navier–Stokes equation gives a deterministic description of the dynamics of a fluid both with respect to the wave function and with respect to velocities. It is shown that taking relativistic effects into account, the Navier–Stokes equation can have a physically meaningful, classical, globally smooth solution of Hubble type, which modifies the isotropic energy-dominance condition, eliminates the cosmological singularity, and accords with the observational data indicating that the Hubble parameter increases with time. The fine structure of the mathematical constants can contain information about interactions of matter. This fact can be used to solve problems on information loss in black holes.