# 2015 DiPerna Lectures

### The 2015 DiPerna Lecture was given by Vladimir Sverak (University of Minnesota) on Thursday, January 22, 2015, 4PM in 60 Evans.

### PDE aspects of 2d incompressible flows

*Abstract : While the mathematical analysis of the PDEs describing the 3d incompressible flows continues to grapple with the notoriously difficult basic questions of existence and uniqueness, the theory of 2d flows is more developed. For the basic equations (Euler and Navier-Stokes) the existence/uniqueness theory has long been established, and more detailed properties of solutions can be studied. Difficult open problems still abound, and recent progress on some of them will be discussed.
In many cases the 2d theory is "critical", on the borderline of what is possible by current methods. Adding simple terms, such as those related to thermal effects (Boussinesq system), can move the equations into the class of problems for which basic existence/uniqueness issues are still open. A recent numerical simulation by G. Luo and T. Hou motivated new developments on this front, and some of the related topics (including interesting 1d models) will also be discussed.*