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

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 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 $t\in [0,T^{\ast })$ with $T^{\ast }\ge 2\nu \cdot \Vert u_0{\Vert }_{\mathsf{A},0}^{-2}$. $\Vert u_0{\Vert }_{\mathsf{A},0}$ 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 $\Vert u_0{\Vert }_{\mathsf{A},0}\le \nu$ (small initial data) we get $T^{\ast }=\infty$. All results hold for all dimensions $n\ge 2$ and are independent on $n$.

extended literature, corrected typos, included uniqueness, slightly rewritten for better reading, new lower time bounds independent on $n$ and independent on first or higher derivatives

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