On the $$L^p$$-semigroups for Stokes equations with dynamic slip boundary conditions in the half-space
Year: 2026
Type: article
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. $$W^{1,p}$$ W 1 , p ) and strong (hence $$W^{2,p}$$ 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 $$L^p$$ 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 $$L^p(\Omega )\hspace{1.111pt}{\times }\hspace{1.111pt}L^p(\partial \Omega )$$ 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 $$\mathcal {H}^{\infty }$$ H ∞ -calculus are implicitly present; but we stay away from the concept of R -boundedness and related heavy functional analytic machinery.
Source: European Journal of Mathematics
Authors: Dalibor Pražák, Michael Zelina
Institutions: Charles University, Uppsala University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,490
OpenAlex ID: W7160868479