The global existence of the smoothing solution for the Navier-Stokes equations

TL;DR

This paper proves the global existence of smoothing solutions for the Navier-Stokes equations by transforming them into linear and quadratic equations, then applying fixed-point theorems.

Contribution

It introduces a novel approach to establish the global existence of solutions by converting Navier-Stokes into linear and quadratic forms and using fixed-point theory.

Findings

Established the explicit general solution for linear equations

Converted Navier-Stokes into linear and quadratic equations

Proved the existence of smoothing solutions globally

Abstract

This paper discussed the global existence of the smoothing solution for 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's equation, the heat-conduct equation, the Schauder fixed-point theorem to prove that the fixed-point is exist, hence the smoothing solution for the Navier-Stokes equations is globally exist.