Publication 13-CNA-017
**Global existence for two extended Navier-Stokes systems**

Mihaela Ignatova

Department of Mathematics

University of California, Riverside

Riverside, California

ignatova@math.ucr.edu

Gautam Iyer

Carnegie Mellon University

Department of Mathematical Sciences

Pittsburgh, PA

gautam@math.cmu.edu

James P. Kelliher

Department of Mathematics

University of California, Riverside

Riverside, California

kelliher@math.ucr.edu

Robert L. Pego

Carnegie Mellon University

Department of Mathematical Sciences

Pittsburgh, PA

rpego@andrew.cmu.edu

Arghir D. Zarnescu

University of Sussex

Falmer, UK

A.Zarnescu@sussex.ac.uk

**Abstract: **We prove global existence of weak solutions to two systems of equations which extend the dynamics of the Navier-Stokes equations for incompressible viscous flow with no-slip boundary condition. The systems of equations we consider arise as formal limits of time discrete pressure-Poisson schemes introduced by Johnston and Liu (J. Comp. Phys. 199 (2004) 221-259) and by Shirokoff and Rosales (J Comp. Phys 230 (2011) 8619-8646) when the initial data does not satisfy the required compatibility condition. Unlike the results of Iyer et al (J. Math. Phys. 53 (2012) 115605), our approach proves existence of weak solutions in domains with less than $C^1$ regularity. Our approach also addresses uniqueness in 2D and higher regularity.

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