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Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system. (English) Zbl 1187.35191

The authors treat here the Cauchy problem for the coupled Dirac-Klein-Gordon (DKG) system on 1+3 :(-iγ μ μ +M)ψ=ϕψ, ( t 2 -Δ+m 2 )ϕ=ψ γ 0 γ, t>0. They study local (in time) well-posedness (LWP) for low regularity data.

Theorem. DKG in 1+3 dimension has LWP for data (ψ 0 ,ϕ 0 ,ϕ 1 )H ε ×H 1/2+ε ×H -1/2+ε for all ε>0. Here f H s =ξ s f ^(ξ) L 2 (ε) , ξ=1+|ξ|. Iteration scheme is used in the proof. Let u H s,b =ξ s |τ|-|ξ| b u ˜(τ,ξ) L 2 (τ,ε) , and f H ˙ s =|ξ| s f ˜(ξ) L 2 (ε) .

The proof relies on the null structure of the system, combined with the bilnear space-time estimates: |D| -s 3 (u(t)v(t)) L 2 (R 1+3 ) C s 1 ,s 2 ,s 3 u 0 H ˙ s 1 ·v 0 H ˙ s 2 , D=/i,

u(t)=exp(±it|D|)u 0 (C):s 1 +s 2 +s 3 =1,s 1 ,s 2 ,s 3 <1,s 1 +s 2 >1/2·

That is, they obtain H s 1 ,b ·H s 2 ,b H -s 3 ,0 for b>1/2 and for s 1 ,s 2 ,s 3 >0 satisfying the condition (C), and prove the estimates of βΠ [±] (D)ψ,Π ± (D)ψ ' for two norms. X·YZ means uv Z Cu X ·v Y for a constant C.

MSC:
35Q40PDEs in connection with quantum mechanics
35Q53KdV-like (Korteweg-de Vries) equations
81Q05Closed and approximate solutions to quantum-mechanical equations
35B65Smoothness and regularity of solutions of PDE