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

TL;DR

This work proves the existence and uniqueness of smooth solutions to the three-dimensional incompressible Navier–Stokes equations by linking them to the parabolic inertial Lamé system. It establishes global well-posedness for the inertial Lamé equations in $\mathbb{R}^3$ with uniform-in-$\lambda$ estimates, and then passes to the limit $\lambda\to\infty$ to obtain a global smooth Navier–Stokes solution $\boldsymbol{u}_\mu$ with divergence-free velocity and corresponding pressure $p_\mu$. Key steps include explicit semigroup representations for linear Lamé problems, construction of the general fundamental solution with sharp bounds, and a contraction-mapping approach for local well-posedness, bootstrapped to global via energy and maximum-principle-type estimates. The results also cover a periodic-boundary variant, providing a robust, constructive route to global smooth NS solutions and suggesting a computational pathway through solving the inertial Lamé system first and then taking a Lamé-parameter limit.

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).