April 21 - June 30, Lectures: Tuesday 15:30 - 17:00
Seminars: tba, ESI, Schrödinger lecture hall.

 
Abstract:
There is an intimate connection between functional inequalities and evolution equations. On the one hand functional inequalities are used to show existence of flows generated by linear or non-linear partial differential equations. Conversely, flows may be used to prove certain inequalities. Liapounov functions, i.e., functions that are monotonic along the flow generate functional inequalities. In this course we study this interplay in a number of examples that have either some physical relevance or are of independent mathematical interest.

We start the course with concentration compactness merely for reviewing measure theory, Sobolev spaces and for fixing the notation. Further topics are: Nash's inequality, Logarithmic Sobolev Inequality for obtaining smoothing estimates for the two dimensional Navier Stokes equations. A nice example for proving sharp inequalities using flows are conformally invariant inequalities such as Sobolev, Hardy-Littlewood-Sobolev inequalities and Hardy-Sobolev-Maz'ya inequalities. Some of these can be understood using transportation theory of which we give a short review. An interesting application of the logarithmic Hardy-Littlewood-Sobolev inequality is the study of the stability of the solutions of the Keller-Segel model, a simple model for describing chemo-taxis.

Transportation theory can also be used to give a proof of Brascamp-Lieb inequalities. Interestingly, these can also be proved using non-linear heat flows. The porous medium equation and its connection to the Gagliardo-Nirenberg inequalities further amplifies the connection between functional inequalities and evolution equations. There is a beautiful connection between transportation theory and porous medium equations that I hope to be able to explain.

Since this lecture is closely linked to the program on Spectral Inequalities at the Erwin Schrödinger Institute, other topics could be considered, such as Lieb-Thirring inequalities and a streamlined proof of the Stability of Matter.