The existence for the classical solution of the Navier-Stokes equations

TL;DR

This paper investigates the existence of classical solutions to the Navier-Stokes equations by transforming them into integral equations and applying Leray-Schauder degree theory within Sobolev spaces.

Contribution

It introduces a method to establish the existence of classical solutions for Navier-Stokes equations using integral transforms and topological degree theory.

Findings

Existence of classical solutions under certain conditions

Application of Leray-Schauder degree in Sobolev spaces

Transformation of Navier-Stokes into generalized integral equations

Abstract

In this paper we will discuss the existence for the classical solution of the Navier-Stokes equations. First, we transform it into generalized integral equations. Next, we discuss the existence of the classical solution by Leray-Schauder degree and Sobolev space\ $H^{-m_{1}}(\Omega_{1})$.