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This note concerns well-posedness of the Stokes and Navier-Stokes equationson uniform $C^{2,1}$-domains on $L_q$. In particular, classes of non-Helmholtzdomains, i.e., domains for which the Helmholtz decomposition does not exist,are adressed. On the one hand, it is proved that the Stokes equations subjectto partial slip in general are not well-posed in the standard setting thatusually applies for Helmholtz domains. On the other hand, it is proved thatunder certain reasonable assumptions the Stokes and Navier-Stokes equationssubject to partial slip are well-posed in a generalized setting. This settingrelies on a generalized version of the Helmholtz decomposition which existsunder suitable conditions on the intersection and the sum of gradient andsolenoidal fields in $L_q$. The proved well-posedness of the Stokes resolventproblem turns even out to be equivalent to the existence of the generalizedHelmholtz decomposition. The presented approach, for instance, includes thesector-like non-Helmholtz domains introduced by Bogovski\u{\i} and Maslennikovaas well as further wide classes of uniform $C^{2,1}$-domains.

Stokes and Navier-Stokes equations subject to partial slip on uniform $C^{2,1}$-domains in $L_q$-spaces