  This note concerns well-posedness of the Stokes and Navier-Stokes equations
on uniform $C^{2,1}$-domains on $L_q$. In particular, classes of non-Helmholtz
domains, i.e., domains for which the Helmholtz decomposition does not exist,
are adressed. On the one hand, it is proved that the Stokes equations subject
to partial slip in general are not well-posed in the standard setting that
usually applies for Helmholtz domains. On the other hand, it is proved that
under certain reasonable assumptions the Stokes and Navier-Stokes equations
subject to partial slip are well-posed in a generalized setting. This setting
relies on a generalized version of the Helmholtz decomposition which exists
under suitable conditions on the intersection and the sum of gradient and
solenoidal fields in $L_q$. The proved well-posedness of the Stokes resolvent
problem turns even out to be equivalent to the existence of the generalized
Helmholtz decomposition. The presented approach, for instance, includes the
sector-like non-Helmholtz domains introduced by Bogovski\u{\i} and Maslennikova
as well as further wide classes of uniform $C^{2,1}$-domains.
