Global solutions for the Dirac-Proca equations with small initial data in 3+1 space time dimensions
type: Item
instance of: scholarly article
title: Global solutions for the Dirac-Proca equations with small initial data in 3+1 space time dimensions
zbMATH Open document ID: 1020.35079
DOI: 10.1016/S0022-247X(02)00662-5
author: Yoshio Tsutsumi
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-25T00:00:00Z
review text: The author treats the Dirac-Proca equations (D-P eqs.): \[ i\gamma^\mu \partial_\mu\psi= \gamma^\mu A_\mu(I- \gamma_5)\psi(t, x), \] \[ \partial_\mu \partial^\mu A^\nu+ M^2A^\nu- \partial^\nu\partial_\mu A^\mu= \langle\gamma^0 \gamma^\nu(I-\gamma_5)\psi,,\psi\rangle/2, \] \(\nu= 0,1,2,3\), \((t,x)\in\mathbb{R}^{1+3}\), with \(\psi(0, x)= \psi_0(x)\in H^{m,m}\), \(A^\nu(0, x)= a^\nu(x)\in H^{m+1,m}\), \(\partial_0 A^\nu(0,x)= b^\nu(x)\in H^{m,m}\), where \(m(\geq 20)\) is an integer. The unique local existence of a smooth solution is standard. Let \(0\leq \varepsilon< 1/4\) and \(\Gamma= (\partial_\mu, \Omega_{\mu j}= x_\mu\partial_j- x_j\partial_\mu)\). Main results: (1) If \((\psi_0,a^\nu, b^\nu)\) satisfies the estimation \(\|\psi_0\|+\|a^\nu\|+\|b^\nu\|\leq\delta\) for a sufficiently small \(\delta> 0\), and if \(\partial_\mu A^\nu= 0\) holds at \(t=0\), then there exists a unique global solution \((\psi,A^\nu)\) of the Cauchy problem of the D-P eqs. (2) \((\psi,A^\nu)\) has a unique free profile \((\psi_{+0},a^\nu_+, b^\nu_+)\) such that \(\{\|\psi- \psi_+\|+\|A^\nu- A^\nu_+\|+ \|\partial_0(A^\nu- A^\nu_+)\|\}(t)\to 0\) as \(t\to\infty\), where \((\psi_+, A^\nu_+)\) are the solutions of the linear D-P eqs. A priori estimates \(\||(\psi, A^\nu)(t)|\|_m\leq C_0\), \(t>0\), with the norm given by \(\Gamma\) prove the results.
reviewed by: Hideo Yamagata
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 35L70
zbMATH DE Number: 1915722
zbMATH Keywords: a priori estimates
zbMATH Keywords: Dirac-Proca equations
zbMATH Keywords: unique global solution
zbMATH Keywords: Cauchy problem
MaRDI profile type: Publication
cites work: Gauge theories in particle physics.
cites work: Q3825451
cites work: Existence of global solutions of the Yang-Mills, Higgs and spinor field equations in $3+1$ dimensions
cites work: Global solutions of nonlinear hyperbolic equations for small initial data
cites work: Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations
cites work: Global solution of the system of wave and Klein-Gordon equations
cites work: Q3992010
cites work: Decay estimates for the Klein ndesh Gordon equation
cites work: Weighted Sobolev spaces applied to nonlinear Klein-Gordon equation
cites work: Q3726637
cites work: Q3331943
cites work: Lectures on nonlinear hyperbolic differential equations
cites work: Small data blow-up for semilinear Klein-Gordon equations
cites work: Uniform decay estimates and the lorentz invariance of the classical wave equation
cites work: Global existence of small amplitude solutions to nonlinear klein-gordon equations in four space-time dimensions
cites work: Q3733338
cites work: The unit condition and global existence for a class of nonlinear Klein- Gordon equations
cites work: Restriction theorems and semilinear Klein-Gordon equations in \((1+3)\)-dimensions
cites work: Normal form and global solutions for the Klein-Gordon-Zakharov equations
cites work: Asymptotic behavior of the solutions of maxwell—klein—gordon field equations in 4—dimensional minkowski space
cites work: Normal forms and quadratic nonlinear Klein-Gordon equations
cites work: Global existence of small amplitude solutions for the Klein-Gordon-Zakharov equations
full work available at URL: s0022-247x(02)00662-5
OpenAlex ID: W1987912809
description: scientific article; zbMATH DE number 1915722