On the Existence and Temporal Asymptotics of Solutions for the Two and Half Dimensional Hall MHD

Abstract

In this paper, we deal with the $$2\frac{1}{2}$$ dimensional Hall MHD by taking the velocity field u and the magnetic field B of the form $$u(t,x,y)=\left( \nabla ^{\perp }\phi (t,x,y), W(t,x,y)\right) $$ and $$B(t,x,y)=\left( \nabla ^{\perp }\psi (t,x,y), Z(t,x,y)\right) $$ . We begin with the Hall equations (without the effect of the fluid part). In this case, we provide several results such as the long time behavior of weak solutions, weak-strong uniqueness, the existence of local and global in time strong solutions, decay rates of $$(\psi ,Z)$$ , the asymptotic profiles of $$(\psi ,Z)$$ , and the perturbation around harmonic functions. In the presence of the fluid field, the results, by comparison, fall short of the previous ones in the absence of the fluid part and we show the existence of local and global in time strong solutions.