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Global solutions for the Dirac-Proca equations with small initial data in 3+1 space time dimensions. (English) Zbl 1020.35079

The author treats the Dirac-Proca equations (D-P eqs.):

iγ μ μ ψ=γ μ A μ (I-γ 5 )ψ(t,x),
μ μ A ν +M 2 A ν - ν μ A μ =γ 0 γ ν (I-γ 5 )ψ,,ψ/2,

ν=0,1,2,3, (t,x) 1+3 , with ψ(0,x)=ψ 0 (x)H m,m , A ν (0,x)=a ν (x)H m+1,m , 0 A ν (0,x)=b ν (x)H m,m , where m(20) is an integer. The unique local existence of a smooth solution is standard. Let 0ε<1/4 and Γ=( μ ,Ω μj =x μ j -x j μ ).

Main results: (1) If (ψ 0 ,a ν ,b ν ) satisfies the estimation ψ 0 +a ν +b ν δ for a sufficiently small δ>0, and if μ A ν =0 holds at t=0, then there exists a unique global solution (ψ,A ν ) of the Cauchy problem of the D-P eqs.

(2) (ψ,A ν ) has a unique free profile (ψ +0 ,a + ν ,b + ν ) such that {ψ-ψ + +A ν -A + ν + 0 (A ν -A + ν )}(t)0 as t, where (ψ + ,A + ν ) are the solutions of the linear D-P eqs.

A priori estimates |(ψ,A ν )(t)| m C 0 , t>0, with the norm given by Γ prove the results.

35Q40PDEs in connection with quantum mechanics
81Q05Closed and approximate solutions to quantum-mechanical equations
35L70Nonlinear second-order hyperbolic equations