Title: Resolvent Estimates for the Stokes Operator

Abstract: This talk is concerned with the study of resolvent estimates and the analyticity of the semigroup in $L^p$ for the Stokes operator. In the case of smooth domains ($C^2$), the resolvent estimates are known to hold for all $1<p\leq \infty$. If the domain is Lipschitz, the estimates were established for a limited range of $p$, depending on the dimension, using the method of layer potentials and a real-variable argument. In this talk, I will present some recent work, joint with Jun Geng, for the case of $C^1$ domains. In particular, I will discuss a key step in the case $p=\infty$, which involves some new estimates that connect the pressure to the gradient of the velocity in the $L^q$ average, but only on scales above certain level.