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We consider a family of initial problems for the Navier-Stokes type equationsgenerated 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. Asthe weights control the order of zero at the infinity with respect to the spacevariables for vectors fields under the consideration, this actually leads toinitial problems over a compact manifold with the singular conic point at theinfinity. We prove that each problem from the family induces Fredholm openinjective mappings on elements of the scales. At the step $1$ of the complex wemay apply the results to the classical Navier-Stokes equations forincompressible viscous fluid.

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