Existence, uniqueness and smoothness of a solution for 3D Navier-Stokes equations with any smooth initial velocity. A priori estimate of this solution

Solutions of the Navier-Stokes and Euler equations with initial conditions for 2D and 3D cases were obtained in the form of converging series, by an analytical iterative method using Fourier and Laplace transforms [28], [29]. In those articles the solutions are infinitely differentiable functions, and for several combinations of parameters numerical results are presented. This article presents a fundamentally new method for solving the Navier-Stokes equations. The article provides a detailed proof of the existence, uniqueness and smoothness of the solution of the Cauchy problem for the 3D Navier-Stokes equations with any smooth initial velocity. A priori estimate of this solution is presented. When the viscosity tends to zero, this proof applies also to the Euler equations.