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. <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$W^{1,p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>) and strong (hence <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$W^{2,p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>W</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>) solutions, which are optimal in terms of the data belonging to an appropriate negative Sobolev or fractional Besov space. In the latter case <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^p(\Omega )\hspace{1.111pt}{\times }\hspace{1.111pt}L^p(\partial \Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="1.111pt" /> <mo>×</mo> <mspace width="1.111pt" /> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Our approach remains elementary in the sense that only the classical Mikhlin multiplier theorem will be used. The methods of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {H}^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-calculus are implicitly present; but we stay away from the concept of R-boundedness and related heavy functional analytic machinery.

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

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2026年5月11日

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Dalibor Pražák,
Michael Zelina

摘要

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 $L^p(\Omega )\hspace{1.111pt}{\times }\hspace{1.111pt}L^p(\partial \Omega )$ $L^{p} (Ω) \times L^{p} (\partial Ω)$ . Our approach remains elementary in the sense that only the classical Mikhlin multiplier theorem will be used. The methods of $\mathcal {H}^{\infty }$ $H^{\infty}$ -calculus are implicitly present; but we stay away from the concept of R-boundedness and related heavy functional analytic machinery.

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

摘要