Mathematicians solve fluid flow equations critical for industrial design
Researchers have cracked a decades-old mathematical problem governing how fluids behave at surfaces—a breakthrough with direct implications for engineering everything from microfluidic devices to industrial pumps. The solution provides the first complete toolkit for predicting fluid behavior under realistic boundary conditions, enabling engineers to design more efficient systems without expensive trial-and-error testing.
Originaltitel: 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.