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 ∂tu = ν∆u + P[v∇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 Arzelà-Ascoli. We gain bounds on the a k 's of u depending on v. These bounds provide the small time existence of a unique smooth solution of the Navier-Stokes equation. They prove that if all first derivatives of the solution are bounded for all times t ∈ [0, T ], then the solution is smooth for all t ∈ [0, T ]. We prove that the Navier-Stokes equation with small initial data, e.g. whena unique smooth solution for all times t ≥ 0. All results hold for all dimensions n ≥ 2.