The Cauchy problem for the Navier-Stokes equations

Abstract

Ch. Fefferman in his works two problems for Navier-Stokes equations are set out: one of them is the Cauchy problem and he considers ¾only those solutions that are infinitely smooth functions are physically meaningful¿. In this article, the author received positive answers for the above problem of Ch. Fefferman. He proved the uniqueness and existence of smooth solutions of the Cauchy problem for the Navier-Stokes equations. The ratio between the pressure P and the kinetic energy density E, previously established by the author. is taken as the basis. As a result of in-depth studies of the Cauchy problem for the Navier-Stokes equations, it is shown that E is a bounded, continuous function that satisfies the Laplace equation and has continuous first-order derivatives with respect to t and all kinds of second derivatives with respect to the spatial variables x and is a regular harmonic function in the space R3. An explicit form of E is found with the help of which the Navier-Stokes equations are reduced to a system of linear parabolic equations and the solutions are written out by the Fourier transform that are infinitely differentiable with respect to t and x. The systems of equations for the curl-vector are found. Proven uniqueness, the existence of infinite smoothness. An estimate is obtained linking the curl-vectors with the Reynolds number.