论文标题
关于椭圆形复合物的Navier-Stokes类型方程的稳定性现象
On the stability phenomenon of the Navier-Stokes type Equations for Elliptic Complexes
论文作者
论文摘要
令$ {\ Mathcal X} $为Riemannian $ n $ dimensional Smooth Compact封闭歧管，$ n \ geq 2 $，$ e^i $是$ \ MATHCAL X $和$ \ {a^i，e^i，e^i \} $的平滑矢量捆绑包，是linial linial linial linear fielater Perner perferter offer ofer tord corpersertors。我们考虑由$ \ {a^i，e^i，e^i \} $相关的操作员方程式在各向异性Hölder空间的规模上，上层$ {\ Mathcal x} \ times [0，t]，带有有限的时间$ t> 0 $。使用差速器的属性$ a^i $和抛物线运算符在此规模的空间中，我们将方程式减少到表单$（i+k）u = f $的非线性fredholm运算符方程，其中$ k $是紧凑的连续运算符。看来，在考虑因素下的每个Banach空间的每个点上，Fréchet衍生品$（i+k）'$在空间中的每个点都不断逆转，地图$（i+k）$在空间中是开放的。
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 Hölder 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 Fréchet 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.
