Let ${\mathcal X}$ be a Riemannian $n$-dimensional smooth compact closed manifold, $n\geq 2$, $E^i$ be smooth vector bundles over $\mathcal X$ and $\{A^i,E^i\}$ be an elliptic differential complex of linear first order operators. We consider the operator equations, induced by the Navier-Stokes type equations associated with $\{A^i,E^i\}$ on the scale of anisotropic Holder spaces over the layer ${\mathcal X} \times [0,T]$ with finite time $T > 0$. Using the properties of the differentials $A^i$ and parabolic operators over this scale of spaces, we reduce the equations to a nonlinear Fredholm operator equation of the form $(I+K) u = f$, where $K$ is a compact continuous operator. It appears that the Frechet derivative $(I+K)'$ is continuously invertible at every point of each Banach space under the consideration and the map $(I+K)$ is open and injective in the space.

中文翻译：

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椭圆复形Navier-Stokes型方程的稳定性现象

令 ${\mathcal X}$ 是一个黎曼 $n$ 维光滑紧致闭流形，$n\geq 2$, $E^i$ 是 $\mathcal X$ 和 $\{A^i 上的光滑向量丛,E^i\}$ 是线性一阶算子的椭圆微分复形。我们考虑由与 $\{A^i,E^i\}$ 相关的 Navier-Stokes 类型方程在层 ${\mathcal X} \times [0, T]$ 有限时间 $T > 0$。使用微分 $A^i$ 和抛物线算子在这个空间尺度上的性质，我们将方程简化为 $(I+K) u = f$ 形式的非线性 Fredholm 算子方程，其中 $K$ 是紧凑连续算子。看起来 Frechet 导数 $(I+K)'