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Publication 12-CNA-003

Coercivity and Stability Results for an Extended Navier-Stokes System

Gautam Iyer
Department of Mathematical Sciences,
Carnegie Mellon University,
Pittsburgh, PA 15213

Robert L. Pego
Department of Mathematical Sciences, Carnegie Mellon University
Pittsburgh, PA 15213

Arghir Zarnescu
Department of Mathematics, University of Sussex
Pevensey III, Falmer, BN1 9QH, United Kingdom

Abstract: In this article we study a system of equations that is known to 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 L2 energy estimate. However, through use of new H1 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.

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