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| We study time local and global well-posedness of the Cauchy problem for a system of semirelativistic equations with quadratic nonlinearities.
The existence of local solutions in $H^s$ with $s > 1/2$ follows easily by the Sobolev embedding $H^s \hookrightarrow L^\infty$. In the case where $0 \le s \le 1/2$, the uniform control by $H^s$ norm breaks down and Strichartz type estimates are not sufficient for a contraction argument unless the uniform control by $H^s$ norm is available. In this talk, we introduce the Fourier restriction method to study the Cauchy problem of the semirelativistic system in $H^s$ with $0 \le s \le 1/2$.
We obtain the existence of local solutions in $H^s$ with $s \ge 0$ by a contraction argument based on a Fourier restriction norm. In addition, under a constraint of nonlinearities, the charge conserves and the solutions in $H^s$ with $s \ge 0$ are shown to extend globally. |
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