论文标题
改善了2D中Maxwell-Klein-Gordon系统的适应性良好结果
Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D
论文作者
论文摘要
在两个空间维度上处理了库仑量规和洛伦兹仪表中的麦克斯韦 - 克莱因 - 戈登系统的局部良好性问题，用于两个空间维度，用于具有最小规律性假设的数据。在$ l^2 $的基于$ l^2 $的sobolev Spaces $ h^s $和$ h^l $中的数据中，电磁字段$ ϕ $和潜在的$ a $。最小的规律性假设是$ s> \ frac {1} {2} $和$ l> \ frac {1} {4} $，它留下了$ \ frac {1} {2} {2} $和$ \ \ \ \ \ frac {1} {1} {4} $的间隙，以尊重$ s_c = 0 c = 0 c = 0。可以减少傅里叶 - lebesgue空间中的数据$ \ wideHat {h}^{s，r} $和$ \ wideHat {h}^{l，r} $ to $ s> \ s> \ frac {21}} {16}} {16}} $要缩放满足$ s_c \ to 1 $，$ l_c \ to 1 $ as $ r \ to 1 $。这里$ \ | f \ | _ {\ wideHat {h}^{s，r}}}：= \ | \langle切ξ\ rangle^s \ tilde {f} \ | _ {l^{l^{r'} _ {°{τξ}}} \，，1 <r \ le 2 \，，，，\，，\，\ frac {1} {1} {1} {r}+\ frac {r}+\ \ \ \ \ \ \ \ \ \ \ \ \ frac {1}} $因此，$ ϕ $以及两个量规$ a $缩小了差距。
The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in $L^2$-based Sobolev spaces $H^s$ and $H^l$ for the electromagnetic field $ϕ$ and the potential $A$, respectively. The minimal regularity assumptions are $s > \frac{1}{2}$ and $l > \frac{1}{4}$ , which leaves a gap of $\frac{1}{2}$ and $\frac{1}{4}$ to the critical regularity with respect to scaling $s_c = l_c =0$ . This gap can be reduced for data in Fourier-Lebesgue spaces $\widehat{H}^{s,r}$ and $\widehat{H}^{l,r}$ to $s> \frac{21}{16}$ and $l > \frac{9}{8}$ for $r$ close to $1$ , whereas the critical exponents with respect to scaling fulfill $s_c \to 1$ , $ l_c \to 1 $ as $r \to 1$ . Here $\|f\|_{\widehat{H}^{s,r}} := \| \langle ξ\rangle^s \tilde{f}\|_{L^{r'}_{τξ}} \, , \, 1 < r \le 2 \, , \, \frac{1}{r}+\frac{1}{r'} = 1 \, . $ Thus the gap is reduced for $ϕ$ as well as $A$ in both gauges.
