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# Communications in Mathematical Sciences

## Volume 13 (2015)

### Number 1

### Global existence for two extended Navier-Stokes systems

Pages: 249 – 267

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n1.a12

#### Authors

#### 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 & Liu [J. Comput. Phys. 199, 221–259, 2004] and by Shirokoff & Rosales [J. Comput. Phys. 230, 8619–8646, 2011] when the initial data does not satisfy the required compatibility condition. Unlike the results of Iyer *et al.* [J. Math. Phys. 53, 115605, 2012], 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.

#### Keywords

Navier-Stokes, numerics, global well-posedness

#### 2010 Mathematics Subject Classification

35Q30, 65M06, 76D05, 76M25