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

Abstract

Let $\mathcal{X}$ be a Riemannian $n$-dimensional smooth compact closed manifold, $n\ge 2$, $E^i$ 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 Hölder 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 Fréchet derivative $(I+K{)}^{\prime }$ 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.