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We prove in this paper some results on the complex and fractional powers of the Stokes operator with slip frictionless boundary conditions involving the stress tensor. This is fundamental and plays an important role in the associated parabolic problem and will be used to prove maximal LpL^{p}-LqL^{q} regularity results for the non-homogeneous Stokes problem.

We prove in this paper some results on the complex and fractional powers of the Stokes operator with slip frictionless boundary conditions involving the stress tensor. This is fundamental and plays an important role in the associated parabolic problem and will be used to prove maximal Lp- Lq regularity results for the non-homogeneous Stokes problem. [ABSTRACT FROM AUTHOR]

In this paper, we prove the analyticity of the semi-group generated by the Stokes operator with a boundary condition involving the pressure. This allows us to obtain weak and strong solutions for the time-dependent Stokes problem with the corresponding boundary condition. [ABSTRACT FROM AUTHOR]

In this article we consider the Stokes problem with Navier-type boundary conditions on a domain $${\Omega}$$ , not necessarily simply connected. Since, under these conditions, the Stokes problem has a non trivial kernel, we also study the solutions lying in the orthogonal of that kernel. We prove the analyticity of several semigroups generated by the Stokes operator considered in different functional spaces. We obtain strong, weak and very weak solutions for the time dependent Stokes problem with the Navier-type boundary condition under different hypotheses on the initial data u and external force f. Then, we study the fractional and pure imaginary powers of several operators related with our Stokes operators. Using the fractional powers, we prove maximal regularity results for the homogeneous Stokes problem. On the other hand, using the boundedness of the pure imaginary powers, we deduce maximal $${L^{p}-L^{q}}$$ regularity for the inhomogeneous Stokes problem. [ABSTRACT FROM AUTHOR]

This paper deals with the time-dependent Stokes problem with Navier boundary conditions on -spaces. We prove the analyticity of the Stokes semigroup with Navier boundary conditions on some spaces to be determined. This analyticity allows us to solve the evolutionary Stokes problem and to obtain weak, strong and very weak solutions. [ABSTRACT FROM AUTHOR]