论文标题
存在定理的定理，用于ersatz Navier-Stokes方程的常规周期性解决方案
Existence theorems for regular spatially periodic solutions to ersatz Navier-Stokes equations
论文作者
论文摘要
$ {\ Mathbb r}^n \ times [0，t] $，$ n \ geq 2 $的Navier-Stokes类型方程的初始问题，在空间周期性设置中具有正时$ t $。首先，我们证明该问题会在$ n \，$ -n \，$ -Dim-dimensional torus $ {\ mathbb t}^n $上诱导开放的注射连续映射。接下来，拒绝证明高阶导数的通用估计值的想法，我们在考虑因素的界限方面对非线性映射获得了非线性映射的高度标准。最后，如果我们考虑包含没有“压力” {}的Navier-Stokes类型方程的版本，我们证明映射是汇总的。这为Navier-Stokes类型方程的这种特定eratz的常规解决方案提供了独特性和存在定理。使用的技术包括通过估计预映射中的所有可能发散序列并匹配渐近性的所有可能发散的序列来证明图像的闭合度。以下事实是必不可少的：i）圆环是一个紧凑的封闭歧管，ii）相应的系统为“局部”。
The initial problem for the Navier-Stokes type equations over ${\mathbb R}^n \times [0,T]$, $n\geq 2$, with a positive time $T$ in the spatially periodic setting is considered. First, we prove that the problem induces an open injective continuous mapping on scales of specially constructed function spaces of Bo\-chner-Sobolev type over the $n\,$-dimensional torus ${\mathbb T}^n$. Next, rejecting the idea of proving a universal a priori estimate for high-order derivatives, we obtain a surjectivity criterion for the non-linear mapping under the considerations in terms of boundedness for its inverse images of precompact sets. Finally, we prove that the mapping is surjective if we consider the versions of the Navier-Stokes type equations containing no `pressure'{}. This gives a uniqueness and existence theorem for regular solutions to this particular ersatz of the Navier-Stokes type equations. The used techniques consist in proving the closedness of the image by estimating all possible divergent sequences in the preimage and matching the asymptotics. The following facts are essential: i) the torus is a compact closed manifold, ii) the corresponding system is `local'.
