2026/07/18 by Huali Zhang, Jie Zhou
#math.AP
In this paper, we study the low-regularity Cauchy problem for the Chern--Simons gauged O(3) sigma model in ℝ1+d (d=1,2) under the Lorenz gauge. For d=1, we establish local well-posedness for initial data (\boldsymbolϕ0,A0)∈ Hs1(ℝ)× Hs1-1(ℝ) with s1>\frac12. This improves the previous result of Jin and Huh \citeHJ by one quarter of a derivative and is almost optimal in view of the scaling-invariant regularities H1/2(ℝ) for the matter field and H-1/2(ℝ) for the gauge field. For d=2, we establish local well-posedness for initial data (\boldsymbolϕ0,A0)∈ Hs2(ℝ2)× Hs2-\frac34(ℝ2) with s2>1. This improves the previous result of Jin and Zhang \citeJZ by one quarter of a derivative and brings the regularity threshold close to the scaling-invariant exponents H1(ℝ2) and H0(ℝ2) for the matter and gauge fields, respectively. The analysis relies on two main ingredients. In two space dimensions, we identify the complete null structure of the derivative nonlinearities, allowing the entire system to be treated within a unified null-form framework. In one space dimension, we establish a direct energy estimate in the function space introduced by Keel and Tao, avoiding the finite-propagation reduction to a small-data problem and enabling the low-regularity iteration for general initial data.