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

ABSTRACT

Let $\mathcal{X}$ be a Riemannian n-dimensional smooth 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 $(I+K{)}^{\mathrm{\prime }}$ is continuously invertible at every point of each Banach space under consideration and the map $(I+K)$ is open and injective in the space.

Keywords:

• Navier-Stokes type equations
• elliptic complexes
• Hölder spaces

AMS Subject Classifications:

• Primary: 76D05
• Secondary: 58J10
• 58J35

Acknowledgements

We thank Prof. N. Tarkhanov for an essential help in the preparation of Section 4.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘BASIS’.