On the Lᵖ-semigroups for Stokes equations with dynamic slip boundary conditions in the half-space

We consider the evolutionary Stokes system, coupled with the so-called dynamic slip boundary condition, in the simple geometry of a d -dimensional half-space. Using the standard technique of Fourier transform in tangential directions, we obtain an explicit formula for the resolvent. We then deduce estimates for both the weak (i.e. W1,p W 1 , p ) and strong (hence W2,p W 2 , p ) solutions, which are optimal in terms of the data belonging to an appropriate negative Sobolev or fractional Besov space. In the latter case Lᵖ L p -integrability of the pressure gradient is included. We allow for solutions with non-zero divergence, thus preparing the way for extensions to general domains. As a by-product, we show that the system generates an analytic semigroup in Lᵖ(Ω )1.111pt× 1.111ptLᵖ(∂ Ω ) L p ( Ω ) × L p ( ∂ Ω ) . Our approach remains elementary in the sense that only the classical Mikhlin multiplier theorem will be used. The methods of H∞ H ∞ -calculus are implicitly present; but we stay away from the concept of R -boundedness and related heavy functional analytic machinery.