On the stability phenomenon of the Navier-Stokes type equations for elliptic complexes

Abstract

ABSTRACT Let be a Riemannian n-dimensional smooth closed manifold, , be smooth vector bundles over and be an elliptic differential complex of linear first order operators. We consider the operator equations, induced by the Navier-Stokes type equations associated with on the scale of anisotropic Hölder spaces over the layer with finite time T>0. Using the properties of the differentials and parabolic operators over this scale of spaces, we reduce the equations to a nonlinear Fredholm operator equation of the form , where K is a compact continuous operator. It appears that the Fréchet derivative is continuously invertible at every point of each Banach space under consideration and the map is open and injective in the space.