# Revista Matemática Iberoamericana

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**Volume 32, Issue 3, 2016, pp. 751–794**

**DOI: 10.4171/RMI/898**

Published online: 2016-10-03

Global wellposedness of the equivariant Chern–Simons–Schrödinger equation

Baoping Liu^{[1]}and Paul Smith

^{[2]}(1) Peking University, Beijing, China

(2) University of California Berkeley, USA

In this article we consider the initial value problem for the $m$-equivariant Chern–Simons–Schrödinger model in two spatial dimensions with coupling parameter $g \in \mathbb R$. This is a covariant NLS type problem that is $L^2$-critical. We prove that at the critical regularity, for any equivariance index $m \in \mathbb Z$, the initial value problem in the defocusing case ($g < 1$) is globally wellposed and the solution scatters. The problem is focusing when $g \geq 1$, and in this case we prove that for equivariance indices $m \in \mathbb Z$, $m \geq 0$, 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 initial data $\phi_0 \in L^2$ is $m$-equivariant and satisfies $\| \phi_0 \|_{L^2}^2 < c_{m, g}$. We also show that $\sqrt{c_{m, g}}$ 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 $c_{m, 1} = 8\pi(m + 1)$.

*Keywords: *Nonlinear Schrödinger equation, Chern–Simons term, global existence, scattering

Liu Baoping, Smith Paul: Global wellposedness of the equivariant Chern–Simons–Schrödinger equation. *Rev. Mat. Iberoam.* 32 (2016), 751-794. doi: 10.4171/RMI/898