Time: 10:00-11:00 May 17th, 2024 (Friday)
Introduction: Zhongwei Shen is a member of the inaugural class of the American Mathematical Society (AMS) Fellows (2012). He was awarded a Centennial Fellowship by the AMS in 1997. Shen held the Ralph E. and Norma L. Edwards Research Professorship in Mathematics from 2007 to 2010 and was awarded a University of Kentucky Research Professorship for 2013-2014. In 2016 he was awarded the title of College of Arts and Sciences Distinguished Professor.
Place: MCM110
Title: Resolvent Estimates for the Stokes Operator
Abstract: This talk is concerned with the study of resolvent estimates in $L^p$ for the Stokes operator. Such estimates play an essential role in the functional analytic approach of Fujita and Kato to the nonlinear Navier-Stokes equations in bounded domains. In the case of smooth domains ($C^2$), the resolvent estimate is well known and holds for all $1<p<\infty$. If the domain is Lipschitz, the estimate was 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 discuss some recent work, joint with Jun Geng, for the case of $C^1$ domains. Starting with the upper half-space, using a perturbation argument, we are able to show that the resolvent estimate holds for all $1<p<\infty$. The case of exterior $C^1$ domains is also studied.
