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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

$\begin{array}{cc}\hfill i\left({\partial }_{t}+\alpha ·\nabla \right)\psi +M\beta \psi & =-\varphi \beta \psi ,\hfill \\ \hfill \left(-{\partial }_{t}^{2}+{\Delta }\right)\varphi +m\varphi & =-〈\beta \psi ,\psi 〉\hfill \end{array}$

with initial data

$\psi \left(0\right)={\psi }_{0},\phantom{\rule{1.em}{0ex}}\varphi \left(0\right)={\varphi }_{0},\phantom{\rule{1.em}{0ex}}{\partial }_{t}\varphi \left(0\right)={\varphi }_{1}·\phantom{\rule{2.em}{0ex}}\left(3\right)$

Here, $\psi :{ℝ}^{1+2}\to {ℂ}^{2}$ is a two-spinor field, and $\varphi :{ℝ}^{1+2}\to ℝ$ is real-valued, $m,M\in ℝ$, $\alpha ·\nabla ={\alpha }^{1}{\partial }_{{x}_{1}}+{\alpha }^{2}{\partial }_{{x}_{2}}$ with ${\alpha }^{1},{\alpha }^{2},\beta$ Hermitian $\left(2×2\right)$-matrices satisfying ${\beta }^{2}={\left({\alpha }^{1}\right)}^{2}={\left({\alpha }^{2}\right)}^{2}=I$, ${\alpha }^{j}\beta =-\beta {\alpha }^{j}$, ${\alpha }^{j}{\alpha }^{k}+{\alpha }^{k}{\alpha }^{j}=2{\delta }_{jk}I$.

The main result states that (1)–(3) is globally well-posed for initial data $\psi \in {L}^{2}\left({ℝ}^{2}\right)$, ${\varphi }_{0}\in {H}^{1/2}\left({ℝ}^{2}\right)$, ${\varphi }_{1}\in {H}^{-1/2}\left({ℝ}^{2}\right)$. The solution satisfies $\psi \in {C}^{0}\left({ℝ}^{+},{L}^{2}\left({ℝ}^{2}\right)\right)$, $\varphi \in {C}^{0}\left({ℝ}^{+},{H}^{1/2}\left({ℝ}^{2}\right)\right)$, ${\partial }_{t}\varphi \in {C}^{0}\left({ℝ}^{+},{H}^{-1/2}\left({ℝ}^{2}\right)\right)$. A second theorem deals with more regular initial data ${\psi }_{0}\in {H}^{s}$, ${\varphi }_{0}\in {H}^{s+1/2}$, ${\varphi }_{1}\in {H}^{s-1/2}$ for $s\ge 0$. In the case of smooth data, there exists a global smooth classical solution.

##### MSC:
 35Q55 NLS-like (nonlinear Schrödinger) equations 35L70 Nonlinear second-order hyperbolic equations 35A01 Existence problems for PDE: global existence, local existence, non-existence 35A02 Uniqueness problems for PDE: global uniqueness, local uniqueness, non-uniqueness