The existence and uniqueness of the smoothing solution of the Navier-Stokes equations

TL;DR

This paper proves the existence and uniqueness of smoothing solutions for the Navier-Stokes equations using linear theory, Fourier transforms, and fixed-point theorems, establishing a rigorous mathematical foundation for these solutions.

Contribution

It introduces a novel approach combining linear PDE theory, Fourier analysis, and fixed-point methods to establish the existence and uniqueness of smoothing solutions for Navier-Stokes equations.

Findings

Existence of smoothing solutions is proven except on a measure-zero set.

Uniqueness of solutions is established under the given conditions.

The approach provides explicit solutions via Fourier transform methods.

Abstract

This paper discussed the existence and uniqueness of the smoothing solution of the Navier-Stokes equations. At first, we construct the theory of the linear equations which is about the unknown four variables functions with constant coefficients. Secondly, we use this theory to convert the Navier-Stokes equations into the simultaneous of the first order linear partial differential equations with constant coefficients and the quadratic equations. Thirdly, we use the Fourier transformation to convert the first order linear partial differential equations with constant coefficients into the linear equations, and we get the explicit general solution of it. At last, we convert the quadratic equations into the integral equations or the question to find the fixed-point of a continuous mapping. We use the theories about the Poisson equation, the heat-conduct equation, the Schauder fixed-point theorem and the contraction mapping principle to prove that the fixed-point is exist and unique except a set whose Lebesgue measure is 0, hence the smoothing solution of the Navier-Stokes equations is also exist and unique except a set whose Lebesgue measure is 0.