An Open Mapping Theorem for the Navier-Stokes Equations

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.