On the Existence and Smoothness of the Navier-Stokes Equation I

TL;DR

This paper establishes a sufficient condition for the existence of smooth solutions to the Navier-Stokes equations, demonstrating solutions with decay at infinity and extending them into the complex domain for positive time.

Contribution

It provides a new sufficient condition ensuring smooth, physically reasonable solutions and constructs a smooth curve of solutions extending into the complex domain.

Findings

Existence of smooth solutions under specified conditions

Solutions exhibit decay properties at infinity

Extension of solutions into the complex domain for positive time

Abstract

In this paper, we give a sufficient condition to guarantee the existence of a smooth solution of the Navier-Stokes Equation with the nice decreasing properties at infinity. In this way, we prove the existence of smooth physically reasonable solutions to the Navier-Stokes problem. Additionally, we show the existence of a smooth curve of entire vector fields of order 2 that extends the solution to the complex domain for positive time.