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

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 $\lambda$ tends to infinity (the other Lamé constant $\mu >0$ is fixed).