The Fredholm Navier-Stokes type equations for the de Rham complex over weighted H\"older spaces

TL;DR

This paper studies a family of Navier-Stokes type equations derived from the de Rham complex on weighted H"older spaces, establishing Fredholm properties and applying results to classical Navier-Stokes equations.

Contribution

It introduces a framework for analyzing Navier-Stokes type equations on weighted anisotropic H"older spaces, proving Fredholm properties and extending to classical Navier-Stokes equations.

Findings

Proves Fredholm properties of the equations on weighted H"older spaces.

Establishes a connection to classical Navier-Stokes equations.

Handles problems over compact manifolds with conic singularities.

Abstract

We consider a family of initial problems for the Navier-Stokes type equations generated by the de Rham complex in ${\mathbb R}^n \times [0,T]$, $n\geq 2$, with a positive time $T$ over a scale weighted anisotropic H\"older spaces. As the weights control the order of zero at the infinity with respect to the space variables for vectors fields under the consideration, this actually leads to initial problems over a compact manifold with the singular conic point at the infinity. We prove that each problem from the family induces Fredholm open injective mappings on elements of the scales. At the step $1$ of the complex we may apply the results to the classical Navier-Stokes equations for incompressible viscous fluid.