Analysis Seminar

We consider the derivative nonlinear Schr \"odinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$-critical with respect to scaling. However, not much is known regarding the well-posendess of the equation below $H^{\frac 12}$. In this talk we prove that this problem is globally well-posed for initial data in the Sobolev spaces $H^s$ for $\frac 1 6\leq s<\frac 12$ under some restriction on the $L^2$ norm. The key ingredient in our argument is proving that ensembles of orbits with $L^2$-equicontinuous initial data remain equicontinuous under evolution. This is joint work with Rowan Killip and Monica Visan.