On the temporal decay for the 2D non-resistive incompressible MHD equations

Califano-Chiuderi \cite{CC} gave the numerical observation that the energy of the MHD equations is dissipated at a rate independent of the ohmic resistivity, which was first proved by \cite{RWXZ}[Ren et al., J. Funct. Anal., 2014] (the initial data near $(0,\vec{e}_1)$, $\vec{e}_1=(1,0)$). Precisely, they showed some explicit decay rates of solutions in $L^2$ norm. So a nature question is whether the obtained decay rates in \cite{RWXZ} is optimal. In this paper, we aim at giving the explicit decay rates of solutions in both $L^2$ norm and $L^\infty$ norm. In particular, our decay rate in terms of $L^2$ norm improves the previous work \cite{RWXZ}.