Markov Selections for the Magnetohydrodynamics and the Hall-Magnetohydrodynamics Systems

Abstract

The magnetohydrodynamics system has found rich applications in applied sciences and has history of intense investigation by researchers including mathematicians. On the other hand, the Hall-magnetohydrodynamics system consists of the former system added by a Hall term which is very singular, and the mathematical development on this system has seen its significant progress only in the last several years. In short, the magnetohydrodynamics system, similarly to the Navier–Stokes equations, is semilinear, while the Hall-magnetohydrodynamics system is quasilinear. In this manuscript, we consider the three-dimensional magnetohydrodynamics, as well as the Hall-magnetohydrodynamics systems, and prove that they have Markov selections. In the case of the magnetohydrodynamics system, we prove furthermore a weak–strong uniqueness result and that any Markov solution has the strong Feller property for regular initial conditions. Consequently, it is deduced that if the magnetohydrodynamics system is well posed starting from one initial data, then it is well posed starting from any initial data, verifying a sharp contrast to the deterministic case in which the well posedness for all time with small initial data is well known. In the case of the Hall-magnetohydrodynamics system, we are also able to prove the weak–strong uniqueness result; however, the proof of strong Feller property seems to break down due to the singularity of the Hall term, about which we elaborate in detail.