  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.
