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

Abstract

Based on the essential connection of the parabolic inertia Lamé 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 Lamé equations and by letting a Lamé constant $λ$ tends to infinity (the other Lamé constant $μ>0$ is fixed).