论文标题
存在定理的定理用于空间周期性解决方案的定期解决方案
Existence Theorems for Regular Spatially Periodic Solutions to the Navier-Stokes Equations
论文作者
论文摘要
我们考虑$ r^{3} \ times [0，t] $的Navier-Stokes方程的初始值问题，并在空间周期性设置中为正时$ t $。在$ r^{3} $上识别具有三维圆环$ t^{3} $功能的周期性矢量值函数，我们证明该问题诱导了Bochner-Sobolev类型的特殊构造功能空间的开放式注射和过滤映射。这为Navier-Stokes方程的常规解决方案提供了独特性和存在定理。我们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药来证明图像的闭合度。
We consider the initial value problem for the Navier-Stokes equations over $R^{3} \times [0,T]$ with a positive time $T$ in the spatially periodic setting. Identifying periodic vector-valued functions on $R^{3}$ with functions on the three-dimensional torus $T^{3}$, we prove that the problem induces an open both injective and surjective mapping of specially constructed function spaces of Bochner-Sobolev type. This gives a uniqueness and existence theorem for regular solutions to the Navier-Stokes equations. Our techniques consist in proving the closedness of the image by estimating all possible divergent sequences in the preimage and matching the asymptotics.
