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 (R^d ) 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 R^d \ W,

i ut + D u = |u|4/(d - 2) u

,        u|Bdry(W) = 0,     (2)

where W is a compact, convex obstacle.

• Category: Applied Math and Analysis
• Duration: 01:34:34
• Date:  April 30, 2012 at 4:25 PM
• Views: 123
• Tags:   seminar, Applied Math And Analysis Seminar