Resolvent estimates in $L^{\infty }$ for the Stokes operator in nonsmooth domains

Abstract We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain $\Omega $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> in $\mathbb{R}^{d}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:math> under the assumptions that $\Omega $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> is $C^{1}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> for $d\ge 3$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:math> and Lipschitz for $d=2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> . As a corollary, it follows that the Stokes operator generates a uniformly bounded analytic semigroup in the spaces of bounded solenoidal functions in $\Omega $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> . The smoothness conditions on $\Omega $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> are sharp. The case of exterior domains with nonsmooth boundaries is also studied. The key step in our proof involves new estimates that connect the pressure to the gradient of the velocity in the $L^{q}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> average, but only on scales above certain level.