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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 ${ℝ}^{1+3}:\left(-i{\gamma }^{\mu }{\partial }_{\mu }+M\right)\psi =\phi \psi$, $\left({\partial }_{t}^{2}-{\Delta }+{m}^{2}\right)\phi ={\psi }^{†}{\gamma }^{0}\gamma$, $t>0$. They study local (in time) well-posedness (LWP) for low regularity data.

Theorem. DKG in $1+3$ dimension has LWP for data $\left({\psi }_{0},{\phi }_{0},{\phi }_{1}\right)\in {H}^{\epsilon }×{H}^{1/2+\epsilon }×{H}^{-1/2+\epsilon }$ for all $\epsilon >0$. Here ${\parallel f\parallel }_{{H}^{s}}={\parallel {〈\xi 〉}^{s}\stackrel{^}{f}\left(\xi \right)\parallel }_{{L}^{2}\left(\epsilon \right)}$, $〈\xi 〉=1+|\xi |$. Iteration scheme is used in the proof. Let ${\parallel u\parallel }_{{H}^{s,b}}{=\parallel 〈\xi 〉}^{s}〈|\tau |-{|\xi |〉}^{b}\stackrel{˜}{u}{\left(\tau ,\xi \right)\parallel }_{{L}^{2}\left(\tau ,\epsilon \right)}$, and ${\parallel f\parallel }_{{\stackrel{˙}{H}}^{s}}={\parallel \phantom{\rule{0.166667em}{0ex}}|\xi |}^{s}\stackrel{˜}{f}{\left(\xi \right)\parallel }_{{L}^{2}\left(\epsilon \right)}$.

The proof relies on the null structure of the system, combined with the bilnear space-time estimates: ${\parallel \phantom{\rule{0.166667em}{0ex}}|D|}^{-{s}_{3}}{\left(u\left(t\right)v\left(t\right)\right)\parallel }_{{L}^{2}\left({R}^{1+3}\right)}\le {C}_{{s}_{1},{s}_{2},{s}_{3}}\parallel {u}_{0}{\parallel }_{{\stackrel{˙}{H}}^{{s}_{1}}}·{\parallel {v}_{0}\parallel }_{{\stackrel{˙}{H}}^{{s}_{2}}}$, $D=\nabla /i$,

$u\left(t\right)=exp\left(±it|D|\right){u}_{0}⇄\left(C\right):{s}_{1}+{s}_{2}+{s}_{3}=1,\phantom{\rule{4pt}{0ex}}{s}_{1},{s}_{2},{s}_{3}<1,\phantom{\rule{4pt}{0ex}}{s}_{1}+{s}_{2}>1/2·$

That is, they obtain ${H}^{{s}_{1},b}·{H}^{{s}_{2},b}\to {H}^{-{s}_{3},0}$ for $b>1/2$ and for ${s}_{1},{s}_{2},{s}_{3}>0$ satisfying the condition $\left(C\right)$, and prove the estimates of $\parallel 〈\beta {{\Pi }}_{\left[±\right]}\left(D\right)\psi ,{{\Pi }}_{±}\left(D\right){\psi }^{\text{'}}〉\parallel$ for two norms. $X·Y\to Z$ means ${\parallel uv\parallel }_{Z}\le {C\parallel u\parallel }_{X}·{\parallel v\parallel }_{Y}$ for a constant $C$.

##### MSC:
 35Q40 PDEs in connection with quantum mechanics 35Q53 KdV-like (Korteweg-de Vries) equations 81Q05 Closed and approximate solutions to quantum-mechanical equations 35B65 Smoothness and regularity of solutions of PDE