An Open Mapping Theorem for the Navier-Stokes Equations

TL;DR

This paper proves an open mapping theorem for the Navier-Stokes equations by transforming them into a nonlinear Fredholm equation and demonstrating the invertibility and openness of the associated operator in anisotropic H"older spaces.

Contribution

It establishes an open mapping theorem for Navier-Stokes equations in weighted H"older spaces, providing new insights into their solution structure.

Findings

The Navier-Stokes equations can be reduced to a nonlinear Fredholm equation.

The operator involved is shown to be invertible and open in the chosen function space.

The results apply to solutions with finite energy estimates over a finite time interval.

Abstract

We consider the Navier-Stokes equations in the layer ${\mathbb R}^n \times [0,T]$ over $\mathbb{R}^n$ with finite $T > 0$. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form $(I+K) u = f$, where $K$ is a compact continuous operator in anisotropic normed H\"older spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all $t \in [0,T]$. On using the particular properties of the de Rham complex we conclude that the Fr\'echet derivative $(I+K)'$ is continuously invertible at each point of the Banach space under consideration and the map $I+K$ is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of H\"older spaces.