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

Abstract

In this paper, we study the wellposedness of the Hall-magnetohydrodynamic system augmented by the effect of electron inertia. Our main result consists of generalizing the wellposedness one in [17] from the Sobolev context to the general Besov spaces and Kato-Herz space. Then, we show that we can reduce the required regularity of the magnetic field in the first result modulo an additional condition on the maximal time of existence. Finally, we show that, for all p∈(3,∞), the Lˆp (and eventually the Lp) norm of the solution (u,B,∇×B), associated to small initial data in Bˆp,∞3p−1(R3), is controlled by t−12(1−3p), which provides a polynomial decay to zero of the Lˆp norm of the solution.