scholarly journals On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge

2014 ◽  
Vol 34 (5) ◽  
pp. 2389-2403 ◽  
Author(s):  
Jianjun Yuan ◽  
Keyword(s):  
Well Posedness ◽  
Lorenz Gauge ◽  
Chern Simons

WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE CHERN–SIMONS–DIRAC SYSTEM IN TWO DIMENSIONS

2013 ◽  
Vol 10 (04) ◽  
pp. 735-771 ◽  
Author(s):  
MAMORU OKAMOTO
Keyword(s):  
Cauchy Problem ◽  
Dirac Equation ◽  
Sobolev Space ◽  
Gauge Invariance ◽  
Two Dimensions ◽  
Well Posedness ◽  
Dirac System ◽  
The Cauchy Problem ◽  
Chern Simons ◽  
Well Posed

We consider the Cauchy problem associated with the Chern–Simons–Dirac system in ℝ1+2. Using gauge invariance, we reduce the Chern–Simons–Dirac system to a Dirac equation and we uncover the null structure of this Dirac equation. Next, relying on null structure estimates, we establish that the Cauchy problem associated with this Dirac equation is locally-in-time well-posed in the Sobolev space Hs for all s > 1/4. Our proof uses modified L4-type estimates.

scholarly journals Global well-posedness of the Chern-Simons-Higgs equations with finite energy

2013 ◽  
Vol 33 (6) ◽  
pp. 2531-2546 ◽  
Author(s):  
Sigmund Selberg ◽  
◽  
Achenef Tesfahun
Keyword(s):  
Finite Energy ◽  
Well Posedness ◽  
Chern Simons

scholarly journals Almost critical local well-posedness for the space-time monopole equation in Lorenz gauge

2015 ◽  
Vol 17 (03) ◽  
pp. 1450043 ◽  
Author(s):  
Achenef Tesfahun
Keyword(s):  
Initial Data ◽  
Critical Exponent ◽  
Sobolev Spaces ◽  
Lebesgue Space ◽  
Space Time ◽  
Well Posedness ◽  
Critical Space ◽  
Lorenz Gauge

Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in Hs with [Formula: see text]. The equation is L2-critical, and hence a [Formula: see text] derivative gap is left between their result and the scaling prediction. In this paper, we consider initial data in the Fourier–Lebesgue space [Formula: see text] for 1 < p ≤ 2 which coincides with Hs when p = 2 but scales like lower regularity Sobolev spaces for 1 < p < 2. In particular, we will see that as p → 1+, the critical exponent [Formula: see text], in which case [Formula: see text] is the critical space. We shall prove almost optimal local well-posedness to the space-time monopole equation in Lorenz gauge with initial data in the aforementioned spaces that correspond to p close to 1.

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