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In this article we consider the Stokes problem with Navier-type boundaryconditions on a domain $\Omega$, not necessarily simply connected. Since underthese conditions the Stokes problem has a non trivial kernel, we also study thesolutions lying in the orthogonal of that kernel. We prove the analyticity ofseveral semigroups generated by the Stokes operator considered in differentfunctional spaces. We obtain strong, weak and very weak solutions for the timedependent Stokes problem with the Navier-type boundary condition underdifferent hypothesis on the initial data $\boldsymbol{u}_0$ and external force$\boldsymbol{f}$. Then, we study the fractional and pure imaginary powers ofseveral operators related with our Stokes operators. Using the fractionalpowers, we prove maximal regularity results for the homogeneous Stokes problem.On the other hand, using the boundedness of the pure imaginary powers we deducemaximal $L^{p}-L^{q}$ regularity for the inhomogeneous Stokes problem.

Semi-group theory for the Stokes operator with Navier-type boundary conditions on $L^{p}$-spaces