In this paper, we analyze new dynamical phenomena of the solutions to the Cauchy problem of Schrödinger equations with Chern–Simons gauge field. This model arises in the non-relativistic gauge theory and captures a strong nonlocal effect due to the presence of the covariant derivative operators. We study the existence and classification of the initial data in \(H^1({\mathbb {R}}^2)\) which generate either global or finite time blow-up solution to the problem under the Coulomb gauge condition, by developing a delicate variational characterization and some suitable scaling properties of the energy functional. In particular, we prove that there exist \(\phi \in H^1({\mathbb {R}}^2)\) and real numbers \(\tau _1 < \tau _2\) , such that the solution u with \(u(0,x) = \tau \phi (\tau x)\) exists globally for any \(\tau < \tau _1\) and blows up in finite time provided \(\tau > \tau _2\) . The Chern–Simons gauge potentials make the problem interesting and different from the classical Schrödinger equations.
