On $L^p$-semigroup to Stokes equation with dynamic slip boundary condition in the half-space

Abstract

We consider 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 the Fourier transform in tangential directions, we obtain an explicit formula for the resolvent. We then deduce estimates for both the weak (i.e. $W^{1,p}$) and strong (hence $W^{2,p}$) solutions, which are optimal in terms of the data belonging to appropriate negative Sobolev or fractional Besov space. In the latter case $L^p$-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 $L^p(\Omega)\times L^p(\partial \Omega)$. Our approach remains elementary in the sense that only the classical Mikhlin multiplier theorem will be used. The methods of $\mathcal{H}^{\infty}$-calculus are implicitly present; but we stay away from the concept of $R$-boundedness and related heavy functional analytic machinery.