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In this paper we study the periodic Navier--Stokes equation. From theperiodic Navier--Stokes equation and the linear equation $\partial_t u =\nu\Delta u + \mathbb{P} [v\nabla u]$ we derive the corresponding equations forthe time dependent Fourier coefficients $a_k(t)$. We prove the existence of aunique smooth solution $u$ of the linear equation by a Montel space version ofArzel\`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 $e^{i\cdot0\cdot x}$ of the initial data $u_0\in C^\infty(\mathbb{T}^n,\mathbb{R}^n)$with $\mathrm{div}\, u_0=0$. For $\|u_0\|_{\mathsf{A},0} \leq \nu$ (smallinitial data) we get $T^* = \infty$. All results hold for all dimensions $n\geq2$ and are independent on $n$.

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