The Cauchy problem for the 3D Navier - Stokes equations. New approach to the solution and its justification

TL;DR

This paper presents a detailed proof of convergence and justification for an analytical iterative method solving the 3D Navier-Stokes equations' Cauchy problem, expanding understanding of solution behavior in a wide parameter range.

Contribution

It introduces a rigorous proof of convergence and justification for an analytical iterative method applied to the 3D Navier-Stokes Cauchy problem, with an estimated convergence boundary.

Findings

Convergence of the iterative method is proven for broad parameter ranges.

An estimated formula for the convergence boundary in parameter space is derived.

The method produces infinitely differentiable solutions satisfying key conditions.

Abstract

Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of solutions for the Navier-Stokes equations in two dimensions have been known for a long time. Leray showed that the Navier-Stokes equations in three space dimensions have a weak solution. Scheffer and Shnirelman obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli, Kohn and Nirenberg improved Scheffer's results, and F.-H. Lin simplified the proof of the results of J. Leray. Many problems and conjectures about behavior of weak solutions of the Euler and Navier-Stokes equations are described in the books of Bertozzi and Majda, Constantin or Lemari\'e-Rieusset. Solutions of the Navier-Stokes and Euler equations with initial conditions (Cauchy problem) for 2D and 3D cases were obtained in the convergence series form by analytical iterative method using Fourier and Laplace transforms in paper $\cite{TT10}$. These solutions were received in a form of infinitely differentiable functions, and that allows us to analyze all aspects of the problem on a much deeper level and with more details. Also such smooth solutions satisfy the conditions required in $\cite{CF06}$ for the problem of Navier-Stokes equations. For several combinations of problem parameters numerical results were obtained and presented as graphs $\cite{TT10}$,$\;\cite{TT11}$. This paper describes detailed proof of convergence of the analitical iterative method for solution of the Cauchy problem for the 3D Navier - Stokes equations. The convergence is shown for wide ranges of the problem's parameters. Estimated formula for the border of convergence area of the iterative process in the space of system parameters is obtained. Also we have provided justification of the analytical iterative method solution for Cauchy problem for the 3D Navier-Stokes equations.