On the stability phenomenon of the Navier-Stokes type Equations for Elliptic Complexes

TL;DR

This paper studies the stability of Navier-Stokes type equations on elliptic complexes over compact manifolds, showing that the associated nonlinear operator equations are well-posed with invertible derivatives.

Contribution

It extends the analysis of Navier-Stokes equations to elliptic complexes on manifolds, demonstrating the invertibility of the Fréchet derivative of the nonlinear operator.

Findings

The nonlinear equations reduce to a Fredholm operator form.

The Fréchet derivative of the operator is invertible everywhere.

The operator map is open and injective in the considered Banach spaces.

Abstract

Let ${\mathcal X}$ be a Riemannian $n$-dimensional smooth compact closed manifold, $n\geq 2$, $E^i$ be smooth vector bundles over $\mathcal X$ and $\{A^i,E^i\}$ be an elliptic differential complex of linear first order operators. We consider the operator equations, induced by the Navier-Stokes type equations associated with $\{A^i,E^i\}$ on the scale of anisotropic H\"older spaces over the layer ${\mathcal X} \times [0,T]$ with finite time $T > 0$. Using the properties of the differentials $A^i$ and parabolic operators over this scale of spaces, we reduce the equations to a nonlinear Fredholm operator equation of the form $(I+K) u = f$, where $K$ is a compact continuous operator. It appears that the Fr\'echet derivative $(I+K)'$ is continuously invertible at every point of each Banach space under the consideration and the map $(I+K)$ is open and injective in the space.