Existence of smooth solutions of the Navier-Stokes equations in three-dimensional Euclidean space

TL;DR

This paper proves the existence of smooth solutions for the three-dimensional incompressible Navier-Stokes equations by linking them to parabolic inertia Lamé equations and analyzing the limit of Lamé constants.

Contribution

It establishes a novel connection between Navier-Stokes and Lamé equations and demonstrates the existence of smooth solutions in 3D Euclidean space.

Findings

Existence and uniqueness of smooth solutions for parabolic inertia Lamé equations.

Smooth solutions of Navier-Stokes obtained as Lamé constant tends to infinity.

Provides a new approach to solving Navier-Stokes equations in 3D.

Abstract

Based on the essential connection of the parabolic inertia Lam\'{e} equations and Navier-Stokes equations, we prove the existence of smooth solutions of the incompressible Navier-Stokes equations in three-dimensional Euclidean space $\mathbb{R}^3$ by showing the existence and uniqueness of smooth solutions of the parabolic inertia Lam\'{e} equations and by letting a Lam\'{e} constant $\lambda$ tends to infinity (the other Lam\'{e} constant $\mu>0$ is fixed).