Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge. Issue 3 (3rd March 2016)

• Record Type: Journal Article
• Authors: Huh, Hyungjin; Oh, Sung-Jin
• Is Part Of: Communications in partial differential equations. Volume 41:Issue 3(2016)
• Page Start: 375
• Page End: 397
• Publication Date: 2016-03-03
• Subjects: Chern-Simons-Higgs -- Chern-Simons-Dirac -- local well-posedness -- null structure -- Lorenz gauge; 35L60 -- 35L70 -- 35Q99; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2015.1132730
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: In this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations in the Lorenz gauge for low regularity initial data. One of our main contributions is the uncovering of a null structure of (CSD). Combined with the standard machinery of X s, b spaces, we obtain local well-posedness of (CSD) for initial data a μ, ψ ∈ H x 1 / 4 + ε . Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local well-posedness for initial data a μ ∈ H x 1 / 4 + ε, ( Φ, ∂ t Φ ) ∈ H x 3 / 4 + ε × H x − 1 / 4 + ε, which improves the previous result of Selberg and Tesfahun.