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Global solutions for the Dirac-Klein-Gordon system in two space dimensions. (English) Zbl 1205.35293

The paper is concerned with the Cauchy problem for the Dirac-Klein-Gordon equations

i( t +α·)ψ+Mβψ=-φβψ,(- t 2 +Δ)φ+mφ=-βψ,ψ

with initial data

ψ(0)=ψ 0 ,φ(0)=φ 0 , t φ(0)=φ 1 ·(3)

Here, ψ: 1+2 2 is a two-spinor field, and φ: 1+2 is real-valued, m,M, α·=α 1 x 1 +α 2 x 2 with α 1 ,α 2 ,β Hermitian (2×2)-matrices satisfying β 2 =(α 1 ) 2 =(α 2 ) 2 =I, α j β=-βα j , α j α k +α k α j =2δ jk I.

The main result states that (1)–(3) is globally well-posed for initial data ψL 2 ( 2 ), φ 0 H 1/2 ( 2 ), φ 1 H -1/2 ( 2 ). The solution satisfies ψC 0 ( + ,L 2 ( 2 )), φC 0 ( + ,H 1/2 ( 2 )), t φC 0 ( + ,H -1/2 ( 2 )). A second theorem deals with more regular initial data ψ 0 H s , φ 0 H s+1/2 , φ 1 H s-1/2 for s0. In the case of smooth data, there exists a global smooth classical solution.


MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
35L70Nonlinear second-order hyperbolic equations
35A01Existence problems for PDE: global existence, local existence, non-existence
35A02Uniqueness problems for PDE: global uniqueness, local uniqueness, non-uniqueness