Abstract:

Let $ \{d_q, \Lambda^{q} \} $ be the 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 the complex with a nonlinear bi-differential operator of zero order $ N^{2} $. Using projection on the next step of the complex, we show that the equation has a unique solution in special Bochner-Sobolev type functional spaces for some (sufficiently small) time $ T^* $.