Thursday, Nov 1: Lenya Ryzhik (Chicago), 4:00 pm

The role of a drift in elliptic and parabolic equations

The first order partial differential equations are closely connected to
the underlying characteristic ODEs. The second order elliptic and
parabolic equations are as closely connected to the Brownian motion and
more general diffusions with a drift. From a variety of points of view,   
the drift does not really matter - all diffusions look more or less
similar. As a caveat to this reasonable line of thought, I will describe a
menagerie of problems: linear and nonlinear, steady and time-dependent,   
compressible and incompressible, where the drift and diffusion confirm the
Mayakovsky thesis "Woe to one alone!" Together, they lead to enhanced
mixing and improved regularity that are impossible for each one of them to
attain.