Local well-posedness for the nonlinear Dirac equation in two space dimensions
type: Item
instance of: scholarly article
title: Local well-posedness for the nonlinear Dirac equation in two space dimensions
zbMATH Open document ID: 1296.35174
author: Hartmut Pecher
published in: Communications on Pure and Applied Analysis
publication date: 2013-11-13T00:00:00Z
reviewed by: Laurent Thomann
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35Q41
Mathematics Subject Classification ID: 35A02
zbMATH DE Number: 6226497
zbMATH Keywords: Dirac equation
zbMATH Keywords: well-posedness
zbMATH Keywords: Fourier restriction norm method
MaRDI profile type: Publication
full work available at URL: cpaa.2014.13.673
OpenAlex ID: W2963301764
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DOI: 10.3934/CPAA.2014.13.673
review text: In this paper, the author considers the Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions, which reads NEWLINE\[NEWLINE\left\{\begin{align*}{ &i(\partial_t +\alpha_1 \partial_{x_1}+\alpha_2 \partial_{x_2})\psi+M\beta \psi=-\langle\beta \psi,\psi \rangle\beta\psi,\;\; (t,x)\in {\mathbb R}\times {\mathbb R}^2, \cr &\psi(0)=\psi_0,\;\; }\end{align*} \right.NEWLINE\]NEWLINE where \(\psi: {\mathbb R}\times {\mathbb R}^2 \longrightarrow {\mathbb C}^2\) is the unknown function, \(M\in {\mathbb R}\) is a fixed constant, and \(\alpha_1,\alpha_2,\beta\) are Hermitian \((2\times 2)\) matrices satisfying some compatibility conditions.NEWLINENEWLINE\quad Using Bourgain-Klainerman-Machedon type estimates, the authors proves that this equation is locally well-posed in \(H^s\) for \(s>1/2\). Then, he also shows that unconditionally uniqueness holds in the space \(C([0,T];H^s({\mathbb R}^2))\) for \(s> 3/4\).
description: scientific article; zbMATH DE number 6226497