4:00 pm Wednesday, February 1, 2012
Geometry-Analysis Seminar: **On the Gross-Pitaevskii Hierarchies**
by **
Natasa Pavlovic** (UT Austin) in HB 227
The Gross-Pitaevskii (GP) hierarchy is an infinite system of coupled linear non-homogeneous PDEs, which appear in the derivation of the nonlinear Schr\"{o}dinger equation (NLS). Inspired by the PDE techniques that have turned out to be useful on the level of the NLS, we realized that, in some instances we can introduce analogous techniques at the level of the GP. In this talk we will discuss some of those techniques which we use to study well-posedness for GP hierarchies. Time permitting, we will also discuss a new derivation of the defocusing cubic GP hierarchy in dimensions $d=2,3$, from an $N$-body Schr\"{o}dinger equation describing a gas of interacting bosons in the GP scaling, in the limit $N\rightarrow\infty$. In particular, we prove convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies, which are inspired by the solutions spaces based on space-time norms introduced by Klainerman and Machedon. We note that in $d=3$, this has been a well-known open problem in the field. The talk is based on joint works with Thomas Chen. Host Department: Rice University-Mathematics Submitted by damanik@rice.edu |