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

Alexander Polkovnikov

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arXiv · 2207.02734

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

Alexander Polkovnikov

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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^* $.

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Alexander Polkovnikov. 2022-07-06. Existence theorem of a weak solution for Navier-Stokes type equations associated with de Rham complex. https://arxiv.org/abs/2207.02734

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