Local well-posedness for the Maxwell-Chern-Simons-Higgs system in Fourier-Lebesgue spaces

Abstract

We consider local well-posedness for the Maxwell-Chern-Simons-Higgs system in Lorenz gauge for data with minimal regularity assumptions in Fourier-Lebesgue spaces ${\hat{H}}^{s,r}$ , where $\Vert u{\Vert }_{{\hat{H}}^{s,r}}:=\Vert ⟨\xi {⟩}^s\hat{u}(\xi ){\Vert }_{L^{r^{\prime }}}$ , and $r$ and $r^{\prime }$ are dual exponents. We show that the gap between this regularity and the regularity with respect to scaling shrinks in the case $r>1$ , $r\to 1$ compared to the classical case $r=2$ .

13 pages. Assumption $s,l,m\ge 1$ in Theorem 1.1 added. arXiv admin note: text overlap with arXiv:2010.06170, arXiv:2012.14239, arXiv:1411.1207, arXiv:1911.05537