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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.

Since the pioneer work of Leray [23] and Hopf [17], Stokes and Navier–Stokes problems have been often studied with Dirichlet boundary condition. Nevertheless, in the opinion of engineers and physicists such a condition is not always realistic in industrial and applied problems of origin. Thus arises naturally the need to carry out a mathematical analysis of these systems with different boundary conditions, which best represent the underlying fluid dynamic phenomenology. Based on the study of the complex and fractional powers of the Stokes operator with pressure boundary condition, we carry out a systematic treatment of the Stokes problem with the corresponding boundary conditions in Lp‐spaces. [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]

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]