  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.
