Existence theorem of a weak solution for Navier-Stokes type equations associated with de Rham complex

Let $\{d_q,{\mathrm{\Lambda }}^q\}$ be de Rham complex on a smooth compact closed manifold $X$ over ${\mathbb{R}}^3$ with Laplacians ${\mathrm{\Delta }}_q$. We consider operator equations, associated with the parabolic differential operators ${\mathrm{\partial }}_t+{\mathrm{\Delta }}_2+N^2$ on the second step of complex with nonlinear bi-differential operator of zero order $N^2$. Using by projection on the next step of complex we show that the equation has unique solution in special Bochner-Sobolev type functional spaces for some (small enough) time $T^{\ast }$.