Global existence for two extended Navier-Stokes systems

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 C¹ regularity. Our approach also addresses uniqueness in 2D and higher regularity.