Irreducibility of the three, and two and a half dimensional Hall-magnetohydrodynamics system

Abstract

The Hall-magnetohydrodynamics system has been considered by many to play an important role in astrophysical plasmas, star formation and magnetic re-connection. The Hall term mathematically elevates the magnetohydrodynamics system from a semi-linear to a quasi-linear type, and standard results which are well-known for the Navier–Stokes equations and the magnetohydrodynamics system, such as the local well-posedness in the non-diffusive case as well as the uniqueness of the weak solution in case the solution depends only on two spatial variables, are completely open for the Hall-magnetohydrodynamics system. Because the Hall-magnetohydrodynamics system is not known to be globally well-posed even if the solution depends only on two spatial variables, a typical method to prove ergodicity for the two-dimensional Navier–Stokes equations, such as Hairer and Mattingly (2006), seems to have no chance. In this manuscript, we take advantage of the structure of the Hall term, follow the work of Flandoli (1997), and prove its irreducibility property. In order to elaborate on the effect of the Hall term, we also prove analogous results for related systems such as the magnetohydrodynamics system, Bénard problem, magnetic Bénard problem, micropolar fluid system, as well as the magneto-micropolar fluid system. At the time of writing this manuscript, due to a technical difficulty, the strong Feller property of the Hall-magnetohydrodynamics system is an open problem.