Global wellposedness of the equivariant Chern-Simons-Schr\"odinger equation

Abstract

In this article we consider the initial value problem for the m-equivariant Chern-Simons-Schr\"odinger model in two spatial dimensions with real-valued coupling parameter g. This is a covariant NLS type problem that is L^2-critical. We prove that at the critical regularity, for any integer-valued equivariance index m, the initial value problem in the defocusing case (g < 1) is globally wellposed and the solution scatters. The problem is focusing when g >= 1, and in this case we prove that for nonnegative integer-valued equivariance indices m there exist constants c = c_{m, g} such that, at the critical regularity, the initial value problem is globally wellposed and the solution scatters when the L^2 initial data phi_0 is m-equivariant and has L^2-norm less than the square root of c_{m, g}. We also show that c_{m, g}^{1/2} is equal to the minimum L^2 norm of a nontrivial m-equivariant standing wave solution. In the self-dual g = 1 case, we have the exact numerical values c_{m, 1} = 8*pi*(m + 1).