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

TL;DR

This paper proves the existence of a weak solution for a Navier-Stokes type equation on a manifold using de Rham complex and functional analysis techniques, extending classical results to a more geometric setting.

Contribution

It establishes a new existence theorem for weak solutions of Navier-Stokes type equations associated with de Rham complexes on manifolds, incorporating nonlinear operators.

Findings

Existence of unique weak solutions in specific functional spaces

Solution valid for small time intervals

Extension of Navier-Stokes theory to geometric complex setting

Abstract

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^* $.