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}.
Categories: math.AP