Local well-posedness for the nonlinear Dirac equation in two space dimensions
In this paper, the author considers the Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions, which reads NEWLINE\[NEWLINE\left\{\begin{align*}{ &i(\partial_t +\alpha_1 \partial_{x_1}+\alpha_2 \partial_{x_2})\psi+M\beta \psi=-\langle\beta \psi,\psi \rangle\beta\psi,\;\; (t,x)\in {\mathbb R}\times {\mathbb R}^2, \cr &\psi(0)=\psi_0,\;\; }\end{align*} \right.NEWLINE\]NEWLINE where \(\psi: {\mathbb R}\times {\mathbb R}^2 \longrightarrow {\mathbb C}^2\) is the unknown function, \(M\in {\mathbb R}\) is a fixed constant, and \(\alpha_1,\alpha_2,\beta\) are Hermitian \((2\times 2)\) matrices satisfying some compatibility conditions.NEWLINENEWLINE\quad Using Bourgain-Klainerman-Machedon type estimates, the authors proves that this equation is locally well-posed in \(H^s\) for \(s>1/2\). Then, he also shows that unconditionally uniqueness holds in the space \(C([0,T];H^s({\mathbb R}^2))\) for \(s> 3/4\).
scientific article; zbMATH DE number 6226497
scientific article