On the $$L^p$$-semigroups for Stokes equations with dynamic slip boundary conditions in the half-space

Abstract 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. $$W^{1,p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:math> ) and strong (hence $$W^{2,p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:math> ) solutions, which are optimal in terms of the data belonging to an appropriate negative Sobolev or fractional Besov space. In the latter case $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -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 $$L^p(\Omega )\hspace{1.111pt}{\times }\hspace{1.111pt}L^p(\partial \Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace/> <mml:mo>×</mml:mo> <mml:mspace/> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Our approach remains elementary in the sense that only the classical Mikhlin multiplier theorem will be used. The methods of $$\mathcal {H}^{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> -calculus are implicitly present; but we stay away from the concept of R -boundedness and related heavy functional analytic machinery.