An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over ${\mathbb R}^n$

We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over ${\mathbb R}^n \times [0,T]$, $n\geq 3$, with a positive time $T$. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations.