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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 $\Bbb R^{1+3}: (-i\gamma^\mu\partial_\mu+ M)\psi= \phi\psi$, $(\partial^2_t- \Delta+ m^2)\phi= \psi^\dag\gamma^0\gamma$, $t> 0$. They study local (in time) well-posedness (LWP) for low regularity data. Theorem. DKG in $1+3$ dimension has LWP for data $(\psi_0,\phi_0,\phi_1)\in H^\varepsilon\times H^{1/2+\varepsilon}\times H^{-1/2+\varepsilon}$ for all $\varepsilon> 0$. Here $\Vert f\Vert_{H^s}= \Vert\langle\xi\rangle^s\widehat f(\xi)\Vert_{L^2(\varepsilon)}$, $\langle\xi\rangle= 1+|\xi|$. Iteration scheme is used in the proof. Let $\Vert u\Vert_{H^{s,b}}= \Vert\langle\xi\rangle^s\langle|\tau|- |\xi|\rangle^b\widetilde u(\tau,\xi)\Vert_{L^2(\tau,\varepsilon)}$, and $\Vert f\Vert_{\dot H^s}= \Vert\,|\xi|^s\widetilde f(\xi)\Vert_{L^2(\varepsilon)}$. The proof relies on the null structure of the system, combined with the bilnear space-time estimates: $\Vert\,|D|^{-s_3}(u(t) v(t))\Vert_{L^2(R^{1+3})}\le C_{s_1,s_2,s_3}\Vert u_0\Vert_{\dot H^{s_1}}\cdot\Vert v_0\Vert_{\dot H^{s_2}}$, $D= \nabla/i$, $$u(t)= \exp(\pm it|D|)u_0\rightleftarrows (C): s_1+ s_2+ s_3= 1,\ s_1,s_2,s_3< 1,\ s_1+ s_2> 1/2.$$ That is, they obtain $H^{s_1,b}\cdot H^{s_2,b}\to H^{-s_3,0}$ for $b> 1/2$ and for $s_1,s_2,s_3> 0$ satisfying the condition $(C)$, and prove the estimates of $\Vert\langle\beta\Pi_{[\pm]}(D)\psi,\Pi_{\pm}(D)\psi'\rangle\Vert$ for two norms. $X\cdot Y\to Z$ means $\Vert uv\Vert_Z\le C\Vert u\Vert_X\cdot\Vert v\Vert_Y$ 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
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