Resolvent estimates for the Stokes operator

Speaker: Shen Zhongwei

Venue: Lecture Hall 210, Haina Complex Building 2, Zijingang Campus

Abstract: The resolvent estimates for the Stokes operator play an essential role in the functional analytic approach of Fujita and Kato to the Navier-Stokes equations. This talk is concerned with the study of resolvent estimates and the analyticity of the semigroup in L^P for the Stokes operator in domains with rough boundaries. In the case of smooth domains (C²), the resolvent estimates are known to hold for all 1< p ≤ ∞. If the domain is Lipschitz, the estimates were established or a limited range of p,depending on the dimension. In this talk, I will present some recent work, joint with Jun Geng, or the case of C¹ domains. In particular, I will discuss a key step in the case p=∞, 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.