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A priori estimates in the energy space for the Chern-Simons-Schrödinger system

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The Chern-Simons-Schrödinger system is a gauge-covariant version of the cubic nonlinear Schrödinger equation in two space dimensions. It describes the effective dynamics of a large system of nonrelativistic charged quantum particles in the plane, interacting with each other and with a self-generated electromagnetic field. In this talk, I will present some recent work towards establishing well-posedness in the energy space H1 for this system. At this regularity, energy methods alone are insufficient 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, Bony’s paraproduct decompositions and the Up and Vp 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.

This talk is part of the Cambridge Analysts' Knowledge Exchange (C.A.K.E.) series.

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