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SUMMARY:A priori estimates in the energy space for the Chern-Simons-Schrö
 dinger system - Zhuo Min &quot\;Harold&quot\; Lim (University of Cambridge
 )
DTSTART:20151125T160000Z
DTEND:20151125T170000Z
UID:TALK61846@talks.cam.ac.uk
CONTACT:Adam Kashlak
DESCRIPTION:The Chern-Simons-Schrödinger system is a gauge-covariant vers
 ion of the cubic nonlinear Schrödinger equation in two space dimensions. 
 It describes the effective dynamics of a large system of nonrelativistic c
 harged quantum particles in the plane\, interacting with each other and wi
 th a self-generated electromagnetic field. In this talk\, I will present s
 ome recent work towards establishing well-posedness in the energy space H^
 1^ for this system. At this regularity\, energy methods alone are insuffic
 ient for proving the required a priori estimates\, and one has to exploit 
 the dispersive nature of the solutions. A major difficulty in doing so is 
 the presence of a nonlinearity involving a derivative\, a part of which is
  not amenable to a perturbative treatment and has to be absorbed into the 
 principal operator. The methods used include Littlewood-Paley theory\, Bon
 y's paraproduct decompositions and the U^p^ and V^p^ spaces introduced by 
 H. Koch and D. Tataru in the context of nonlinear dispersive equations. I 
 will give a gentle\, but necessarily brisk\, overview of these techniques.
LOCATION:MR14\, Centre for Mathematical Sciences
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