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

Let $ \{ d_q, \Lambda^{ q} \} $ be de Rham complex on a smooth compact closed manifold $X$ over $ \mathbb{ R}^3 $ with Laplacians $\Delta_{ q} $. We consider operator equations, associated with the parabolic differential operators $\partial_t + \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^* $.

Categories: math.AP