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

Philipp J. di Dio

Published 2020-10-12Version 1

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 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 $v$. These bounds provide the small time existence of a smooth solution of the Navier--Stokes equation. They prove that if all first derivatives of the solutions are bounded for all times $t\in [0,T]$, then the solutions are smooth for all $t\in [0,T]$. We prove that the Navier--Stokes equation with small initial data, e.g.\ when $\|u_0\|_\mathsf{A} + \sqrt{n}\cdot (\|\partial_1 u_0\|_\mathsf{A} + \dots + \|\partial_n u_0\|_\mathsf{A})\leq \nu$, has a smooth solution for all times $t\geq 0$. All results hold for all dimensions $n\geq 2$.