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Local well-posedness for Chern-Simons gauged $O(3)$ sigma equations under the Lorenz gauge

Jin Guanghui, Huali Zhang

TL;DR

This paper establishes local well-posedness for the Chern-Simons gauged $O(3)$ sigma model under the Lorenz gauge in two spatial dimensions for data near the energy-critical regularity of the matter field. The authors exploit the hidden null-form structure in the nonlinearities and develop bilinear estimates in wave-Sobolev spaces $H^{s,b}$ to control the dynamics, together with a decomposition of the gauge field into divergence-free and curl-free components. The main result shows existence, uniqueness, and continuous dependence for initial data in $H^{s}(\mathbb{R}^2) \times H^{s-\frac12}(\mathbb{R}^2)$ with $s>1$, along with precise regularity for the gauge field in $H^{s-\frac12}$ and $H^{s-\frac32}$ for the temporal derivative. An energy-conservation identity is provided in the appendix, anchoring the local theory to the model's physical invariants. Overall, the work advances low-regularity well-posedness for gauge-coupled nonlinear sigma models and highlights the role of null structure in such systems.

Abstract

In this paper, we study the Cauchy problem for the Chern-Simons gauged $O(3)$ sigma model under the Lorenz gauge condition. We prove the local well-posedness of solutions if the initial matter field and gauge field satisfy $(\bmφ_0, \bA_0) \in H^s(\R^2)\times H^{s-\frac12}(\R^2)$, $s>1$, where the critical regularity for $\bmφ_0$ is $s_c=1$. Our proof is based on identifying null forms within the system and utilizing bilinear estimates in wave-Sobolev space.

Local well-posedness for Chern-Simons gauged $O(3)$ sigma equations under the Lorenz gauge

TL;DR

This paper establishes local well-posedness for the Chern-Simons gauged sigma model under the Lorenz gauge in two spatial dimensions for data near the energy-critical regularity of the matter field. The authors exploit the hidden null-form structure in the nonlinearities and develop bilinear estimates in wave-Sobolev spaces to control the dynamics, together with a decomposition of the gauge field into divergence-free and curl-free components. The main result shows existence, uniqueness, and continuous dependence for initial data in with , along with precise regularity for the gauge field in and for the temporal derivative. An energy-conservation identity is provided in the appendix, anchoring the local theory to the model's physical invariants. Overall, the work advances low-regularity well-posedness for gauge-coupled nonlinear sigma models and highlights the role of null structure in such systems.

Abstract

In this paper, we study the Cauchy problem for the Chern-Simons gauged sigma model under the Lorenz gauge condition. We prove the local well-posedness of solutions if the initial matter field and gauge field satisfy , , where the critical regularity for is . Our proof is based on identifying null forms within the system and utilizing bilinear estimates in wave-Sobolev space.

Paper Structure

This paper contains 13 sections, 6 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Some fundamental properties of CSS
  3. Historical results
  4. Statement of results
  5. Notations
  6. Organization of the rest paper
  7. Preliminaries
  8. Bilinear estimates
  9. Null structure
  10. Proof of Theorem \ref{['mainthm']}
  11. A Prior Energy Estimates.
  12. Local Existence with finite energy.
  13. appendix: Energy conservation

Key Result

Theorem 1.1

Let $s>1$, consider the Cauchy problem of Chern-Simons gauged $O(3)$ sigma equations f0-lorenz, with the initial data in the following Sobolev space: satisfying the constraint constraint1 and $\langle\bm{\phi}_0,\,\bm{\phi}_1\rangle=0$, where Then there exists a unique solution $\mathbf A$ and $\bm{\phi}$ to f0-f3 on $[0,T]\times \mathbb R^2$$(T>0$ and $T$ depends on $\|\mathbf a\|_{H^{s-\frac{1

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1: DFS, Theorem 5.1
  • Lemma 2.2: Selberg,Theorem 12, Theorem 13
  • Remark 2.1
  • Lemma 2.3: FK, Corollary 13.3
  • Lemma 2.4: FK, Corollary 13.4
  • Lemma 2.5: FK, Corollary 13.5