Null structure and almost optimal local well-posedness of the Maxwell-Dirac system

2010 ◽  
Vol 132 (3) ◽  
pp. 771-839 ◽  
Author(s):  
Piero D'Ancona ◽  
Damiano Foschi ◽  
Sigmund Selberg
Keyword(s):  
Well Posedness ◽  
Dirac System

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.

Corrigendum: Local well-posedness of the coupled Yang–Mills and Dirac system for low regularity data (2022 Nonlinearity 35 1)

Nonlinearity ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. C3-C4
Author(s):  
Hartmut Pecher
Keyword(s):  
Well Posedness ◽  
Dirac System ◽  
Low Regularity ◽  
Yang Mills

Abstract An error in the proof of the main theorem is fixed.

scholarly journals Local well-posedness of the coupled Yang–Mills and Dirac system for low regularity data

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 1-29
Author(s):  
Hartmut Pecher
Keyword(s):  
Dirac Equation ◽  
Well Posedness ◽  
Dirac System ◽  
Null Condition ◽  
Low Regularity ◽  
Yang Mills ◽  
Smooth Data

Abstract We consider the classical Yang–Mills system coupled with a Dirac equation in 3 + 1 dimensions. Using that most of the nonlinear terms fulfill a null condition we prove local well-posedness for data with minimal regularity assumptions. This problem for smooth data was solved forty years ago by Choquet-Bruhat and Christodoulou. Our result generalises a similar result for the Yang–Mills equation by Selberg and Tesfahun.

scholarly journals Ill-posedness of the Maxwell–Dirac system below charge in space dimension three and lower

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Sigmund Selberg ◽  
Achenef Tesfahun
Keyword(s):  
Electromagnetic Field ◽  
Space Dimension ◽  
Local Solution ◽  
Well Posedness ◽  
Dirac System ◽  
Smooth Solutions ◽  
Critical Regularity ◽  
Reasonable Sense

AbstractThe Maxwell–Dirac system describes the interaction of an electron with its self-induced electromagnetic field. In space dimension $$d=3$$ d = 3 the system is charge-critical, that is, $$L^2$$ L 2 -critical for the spinor with respect to scaling, and local well-posedness is known almost down to the critical regularity. In the charge-subcritical dimensions $$d=1,2$$ d = 1 , 2 , global well-posedness is known in the charge class. Here we prove that these results are sharp (or almost sharp, if $$d=3$$ d = 3 ), by demonstrating ill-posedness below the charge regularity. In fact, for $$d \le 3$$ d ≤ 3 we exhibit a spinor datum belonging to $$H^s(\mathbb {R}^d)$$ H s ( R d ) for $$s<0$$ s < 0 , and to $$L^p(\mathbb {R}^d)$$ L p ( R d ) for $$1 \le p < 2$$ 1 ≤ p < 2 , but not to $$L^2(\mathbb {R}^d)$$ L 2 ( R d ) , which does not admit any local solution that can be approximated by smooth solutions in a reasonable sense.

Well-posedness for the dimension-reduced Chern–Simons–Dirac system

2016 ◽  
Vol 17 (3) ◽  
pp. 1031-1048 ◽  
Author(s):  
Shuji Machihara ◽  
Mamoru Okamoto
Keyword(s):  
Well Posedness ◽  
Dirac System ◽  
Chern Simons

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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