Timestepping schemes for the 3d Navier-Stokes equations: small solutions and short times

TL;DR

This paper analyzes two time-stepping schemes for the 3d Navier-Stokes equations, showing that solutions remain bounded under small initial data, forcing, or short times, similar to the continuous case.

Contribution

It proves boundedness of semi-implicit and fully implicit discretizations of 3d Navier-Stokes under smallness conditions, extending continuous results to discrete schemes.

Findings

Solutions remain bounded in $H^1$ under small data conditions

Boundedness holds for both semi-implicit and fully implicit schemes

Results apply to short times and small initial data or forcing

Abstract

It is well known that the solution of the 3d Navier--Stokes equations remains bounded if the initial data and the forcing are sufficiently small relative to the viscosity, and for a finite time given any bounded initial data. In this article, we consider two temporal discretisations (semi-implicit and fully implicit) of the 3d Navier--Stokes equations in a periodic domain and prove that their solutions remain bounded in $H^1$ subject to essentially the same smallness conditions (on initial data, forcing or time) as the continuous system and to suitable timestep restrictions.