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

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