18.158 - Topics in Differential Equations (Fall 2011)


Instructor: Gigliola Staffilani

Office: Room 2-246

Email: gigliola [at] math . mit .edu

Lectures: T R 11:00 - 12:30 Room 2-102

Office Hours: TBA

Other: Please see Below


The main object of study in this course will be periodic non linear Schrodinger equations (NLS) as infinite dimensional Hamiltonian systems.

In the first part of this course I will start by introducing the conjecture posed by Bourgain concerning periodic Strichartz estimates on rational tori. I will then present the parts of the conjecture that are proved and I will explain why these results are still not available for irrational tori. I will use the known Strichartz estimates to then prove local and, when possible, global well-posedness results for certain NLS equations defined in rough Sobolev spaces. I will also show that in some cases we can view these global rough solutions as well defined flows of certain infinite dimensional Hamiltonian systems with a symplectic structure. Here we prove that theorems such as Gromov's non squeezing theorem are still available. To finish this part I will then introduce the concepts of weak turbulence and forward cascade and show how these are linked to certain polynomial in time bounds for Sobolev norms of global solutions to NLS equations. In particular I will consider in this context the periodic, cubic, defocusing NLS. The interesting question here is to show that for certain small data such a polynomial growth actually occurs. This is still far from being proved, but I will present a recent result where a certain very weak growth can be indeed obtained.

In the second part of the course I will introduce the concept of Gibbs measure for certain periodic NLS equations in Hamiltonian form. I will present the theorem of Bourgain for certain periodic, focusing and defocusing, one dimensional NLS in which invariance of the Gibbs measure is proved and used to then show almost surely (w.r.t. the Gibbs measure) global well-posedness at a very rough regime of regularity of the initial data. I will continue with the proof of a similar result for the derivative NLS equation, a proof that will force us to introduce a certain gauge transformation and then relate it to the Gibbs measure.

In the final part of the course I will explain how certain NLS equations are linked to the dynamics of the Bose-Einstein condensate. Here I will introduce the periodic Gross-Pitaevskii system and I will show the uniqueness of its solution in rational tori of dimensions 1 and 2 and also in dimension 3 when a certain a priori bound is assumed.

Prerequisites: Knowledge of basic harmonic and Fourier analysis, theory of distributions, basic theory of PDE.

Text Book: Lecture notes, research papers to be announced.


[09.01.2011]  Welcome to the fall semester!