A Tamed 3D Navier-Stokes Equation in Domains

TL;DR

This paper introduces a tamed 3D Navier-Stokes equation in various domains, proving existence, uniqueness, and the presence of a global attractor, thereby extending classical results to more general settings.

Contribution

It establishes the existence and uniqueness of strong solutions for the tamed 3D Navier-Stokes equation in unbounded and bounded domains, and demonstrates the existence of a global attractor.

Findings

Existence and uniqueness of strong solutions

Global attractor in bounded domains

Extension of classical results to unbounded domains

Abstract

In this paper, we analyze a tamed 3D Navier-Stokes equation in uniform $C^2$-domains (not necessarily bounded), which obeys the scaling invariance principle, and prove the existence and uniqueness of strong solutions to this tamed equation. In particular, if there exists a bounded solution to the classical 3D Navier-Stokes equation, then this solution satisfies our tamed equation. Moreover, the existence of a global attractor for the tamed equation in bounded domains is also proved. As simple applications, some well known results for the classical Navier-Stokes equations in unbounded domains are covered.