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

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).