论文标题
麦克斯韦 - 迪拉克系统的不良适应性在空间维度下方的电荷以下是三分之一和较低的
Ill-posedness for the Maxwell-Dirac system below charge in space dimension three and lower
论文作者
论文摘要
Maxwell-Dirac系统描述了电子与其自诱导的电磁场的相互作用。在太空维度$ d = 3 $中，系统是批判性的，也就是说，对于缩放方案而言，旋转器的$ l^2 $ - 重要的是，局部适应性良好几乎归结为关键的规律性。在收费临界尺寸$ d = 1,2 $中，全球适应性良好在电荷类别中是众所周知的。在这里，我们证明这些结果是敏锐的（或者$ d = 3 $），通过证明低于充电规律性的不良性。实际上，对于$ d \ le 3 $，我们展示了一个属于$ h^s（\ mathbb r^d）$的旋转基准，$ s <0 $，以及$ 1 \ le p <2 $的$ s <0 $（\ mathbb r^d）$，但不能传递任何解决方案的求解，但不能传递$ 1 \ le p <2 $，但不适合$ l^2（\ mathbb r^d）$。
The Maxwell-Dirac system describes the interaction of an electron with its self-induced electromagnetic field. In space dimension $d=3$ the system is charge-critical, that is, $L^2$-critical for the spinor with respect to scaling, and local well-posedness is known almost down to the critical regularity. In the charge-subcritical dimensions $d=1,2$, global well-posedness is known in the charge class. Here we prove that these results are sharp (or almost sharp, if $d=3$), by demonstrating ill-posedness below the charge regularity. In fact, for $d \le 3$ we exhibit a spinor datum belonging to $H^s(\mathbb R^d)$ for $s<0$, and to $L^p(\mathbb R^d)$ for $1 \le p < 2$, but not to $L^2(\mathbb R^d)$, which does not admit any local solution that can be approximated by smooth solutions in a reasonable sense.
