Energy Solution to the Chern-Simons-Schrödinger Equations

Abstract

We prove that the Chern-Simons-Schrödinger system, under the condition of a Coulomb gauge, has a unique local-in-time solution in the energy space $H^1({ℝ}^2)$. The Coulomb gauge provides elliptic features for gauge fields $A_0,A_j$. The Koch- and Tzvetkov-type Strichartz estimate is applied with Hardy-Littlewood-Sobolev and Wente's inequalities.