Harmonic Analysis and PDE: Fluid Mechanics and Kato's Problem
The main goal of the courses is to offer to graduate students and interested researchers introductory expositions to two topics in PDE very much related to singular integrals of Calderón-Zygmund type. On one hand Diego Córdoba will explain the formulation of Euler and Navier-Stokes equations from basic principles, will show the use on singular integrals in their study and will then cover some topics of more recent interest: vortex patches and interface problems. On the other hand, Michael Lacey will explain the solution of the Kato problem on the domain of square roots of accretive operators. In dimension one this is equivalent to L2 estimates for the Cauchy integral on Lipschitz graphs and in higher dimensions the proof depends on the T(b) Theorem as shown by Auscher, Lacey, McIntosh, Hofmann and Tchamichan.