Global solutions for the Dirac-Proca equations with small initial data in 3+1 space time dimensions
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.
scientific article; zbMATH DE number 1915722
scientific article