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

2014/10/13 by Youngjoon Hong, Hong, Youngjoon, D. Wirosoetisno +1
Computer Science · Engineering · Mathematics · #65M12 #76D05 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Navier-Stokes equation solutions #Numerical Analysis (math.NA) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1410.3415

openalex publication_date 2014/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 H¹ subject to essentially the same smallness conditions (on initial data, forcing or time) as the continuous system and to suitable timestep restrictions.

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