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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 $$\align i(\partial_t+\alpha\cdot\nabla)\psi+M\beta\psi&= -\varphi\beta\psi, \tag1\\ (-\partial^2_t+\Delta)\varphi+m\varphi&= -\langle\beta\psi,\psi\rangle \tag2 \endalign$$ with initial data $$\psi(0)=\psi_0,\quad \varphi(0)=\varphi_0,\quad \partial_t\varphi(0)=\varphi_1. \tag 3$$ Here, $\psi:\Bbb R^{1+2}\to\mathbb{C}^2$ is a two-spinor field, and $\varphi:\Bbb R^{1+2}\to\Bbb R$ is real-valued, $m,M\in\Bbb R$, $\alpha\cdot\nabla=\alpha^1\partial_{x_1}+\alpha^2\partial_{x_2}$ with $\alpha^1, \alpha^2, \beta$ Hermitian $(2\times 2)$-matrices satisfying $\beta^2=(\alpha^1)^2=(\alpha^2)^2=I$, $\alpha^j\beta=-\beta\alpha^j$, $\alpha^j\alpha^k+\alpha^k\alpha^j=2\delta_{jk}I$. The main result states that (1)--(3) is globally well-posed for initial data $\psi\in L^2(\Bbb R^2)$, $\varphi_0\in H^{1/2}(\Bbb R^2)$, $\varphi_1\in H^{-1/2}(\Bbb R^2)$. The solution satisfies $\psi\in C^0(\Bbb R^+,L^2(\Bbb R^2))$, $\varphi\in C^0(\Bbb R^+, H^{1/2}(\Bbb R^2))$, $\partial_t\varphi\in C^0(\Bbb R^+, H^{-1/2}(\Bbb R^2))$. 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 $s\geq 0$. In the case of smooth data, there exists a global smooth classical solution.

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
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