Coercivity and stability results for an extended Navier-Stokes system

TL;DR

This paper investigates an extended Navier-Stokes system that allows for non-divergence-free velocity fields, establishing new energy estimates and proving global existence and stability results, especially in two dimensions with divergence damping.

Contribution

It introduces novel $H^1$ coercivity estimates for the extended system, enabling new global existence and stability results, including for small divergence and divergence damping.

Findings

Established global existence in 2D with divergence damping.

Developed new $H^1$ coercivity estimates for the linear equations.

Proved stability results for small divergence and time-discrete schemes.

Abstract

In this article we study a system of equations that is known to {\em extend} Navier-Stokes dynamics in a well-posed manner to velocity fields that are not necessarily divergence-free. Our aim is to contribute to an understanding of the role of divergence and pressure in developing energy estimates capable of controlling the nonlinear terms. We address questions of global existence and stability in bounded domains with no-slip boundary conditions. Even in two space dimensions, global existence is open in general, and remains so, primarily due to the lack of a self-contained $L^2$ energy estimate. However, through use of new $H^1$ coercivity estimates for the linear equations, we establish a number of global existence and stability results, including results for small divergence and a time-discrete scheme. We also prove global existence in 2D for any initial data, provided sufficient divergence damping is included.