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Benjamin Dodson : Concentration compactness for the L^2 critical nonlinear Schrodinger equation

The nonlinear Schrodinger equation

i ut + D u = m |u|(4/d)u     (1)
is said to be mass critical since the scaling u(t,x)=l-d/2 u(t/l2 , x/l) preserves the L2 - norm, m = ± 1. In this talk we will discuss the concentration compactness method, which is used to prove global well - posedness and scattering for (1) for all initial data u(0) in L2 (Rd ) when m = +1, and for u(0) having L2 norm below the ground state when m = -1. This result is sharp.

As time permits the talk will also discuss the energy - critical problem in Rd \ W,

i ut + D u = |u|4/(d - 2) u ,        u|Bdry(W) = 0,     (2)
where W is a compact, convex obstacle.

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