Well-posedness and long time behavior for the Electron Inertial Hall-MHD system in Besov and Kato-Herz spaces

In this paper, we study the wellposedeness of the Hall-magnetohydrodynamic system augmented by the effect of electron inertia. Our main result consists of generalising the wellposedness one in \cite{Zhao} from the Sobolev context to the general Besov spaces and Kato-Herz space, then we show that we can reduce the requared regularity of the magnetic field in the first result modulo an additional condition on the maximal time of existence. Finally, we show that the ${\widehat{L}}^p$ (and eventually the $L^p$) norm of the solution $(u,B,\mathrm{\nabla }\times B)$ associated to an initial data in ${\widehat{B}}_{p,\mathrm{\infty }}^{\frac{3}{p}-1}({\mathbb{R}}^3)$, is controled by $t^{-\frac{1}{2}(1-\frac{3}{p})}$, for all $p\in (3,\mathrm{\infty })$, which provides a polynomial decay to zero of the ${\widehat{L}}^p$ norm of the solution.