论文标题
在周期性的Navier中 - Stokes方程：所有维度的基本方法和平滑度$ n \ geq 2 $
On the periodic Navier--Stokes equation: An elementary approach to existence and smoothness for all dimensions $n\geq 2$
论文作者
论文摘要
在本文中，我们研究了周期性的Navier-Stokes方程。从周期性的navier-stokes方程和线性方程$ \ partial_t u =νΔU + \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ nabla u] $我们在时间依赖的傅立叶系数$ a_k（t）$中得出相应的方程。我们证明了通过Arzelà-ascoli的Montel Space版本的线性方程的独特平滑解决方案$ U $。根据$ v $，我们会在$ u $的$ a_k $上获得界限。使用$ v = -u $，这些边界表明，$ n $ dimensional的定期navier的独特平滑解决方案$ u $ in [0，t^*）$带有$ t^* \ geq2ν\ geq2ν\ cdot \ cdot \ cdot \ cdot \ cdot \ | u_0 $ \ | U_0 \ | _ {\ MATHSF {a}，0} $是傅立叶系数的$ l^2 $ -norms的总和，而无需$ e^{i \ cdot 0 \ cdot 0 \ cdot x} $的初始数据$ u_0 \ in C^\ infty（in Inftty）（ $ \ mathrm {div} \，u_0 = 0 $。对于$ \ | | u_0 \ | _ {\ Mathsf {a}，0} \leqν$（小初始数据），我们得到$ t^* = \ infty $。所有尺寸的所有结果均可$ n \ geq 2 $，并且在$ n $上是独立的。
In this paper we study the periodic Navier--Stokes equation. From the periodic Navier--Stokes equation and the linear equation $\partial_t u = νΔu + \mathbb{P} [v\nabla u]$ we derive the corresponding equations for the time dependent Fourier coefficients $a_k(t)$. We prove the existence of a unique smooth solution $u$ of the linear equation by a Montel space version of Arzelà--Ascoli. We gain bounds on the $a_k$'s of $u$ depending on $v$. With $v = -u$ these bounds show that a unique smooth solution $u$ of the $n$-dimensional periodic Navier--Stokes equation exists for all $t\in [0,T^*)$ with $T^* \geq 2ν\cdot \|u_0\|_{\mathsf{A},0}^{-2}$. $\|u_0\|_{\mathsf{A},0}$ is the sum of the $l^2$-norms of the Fourier coefficients without $e^{i\cdot 0\cdot x}$ of the initial data $u_0\in C^\infty(\mathbb{T}^n,\mathbb{R}^n)$ with $\mathrm{div}\, u_0=0$. For $\|u_0\|_{\mathsf{A},0} \leq ν$ (small initial data) we get $T^* = \infty$. All results hold for all dimensions $n\geq 2$ and are independent on $n$.
