On the periodic Navier--Stokes equation: An elementary approach to existence and smoothness for all dimensions $n\geq 2$

Abstract

In this paper we study the periodic Navier--Stokes equation. From the periodic Navier--Stokes equation and the linear equation $\partial_t u = \nu\Delta u + \mathbb{P} [v\nabla u]$ 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 Arzel\`a--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 $T^* \geq 2\nu\cdot \|u_0\|_{\mathsf{A},0}^{-2}$. $\|u_0\|_{\mathsf{A},0}$ 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 $\|u_0\|_{\mathsf{A},0} \leq \nu$ (small initial data) we get $T^* = \infty$. All results hold for all dimensions $n\geq 2$ and are independent on $n$.