On the Temporal Decay for the 2D Non-resistive Incompressible MHD Equations

Califano–Chiuderi (Phys Rev E 60(Part B): 4701–4707, 1999) 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 Ren et al. (J Funct Anal 267: 503–541, 2014) (the initial data near ( 0 , e → 1 ) , 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 Ren et al. (J Funct Anal 267: 503–541, 2014) are optimal. In this paper, we aim at giving the explicit decay rates of solutions in both L 2 norm and L ∞ norm. In particular, our decay rate in terms of L 2 norm improves the previous work Ren et al. (J Funct Anal 267: 503–541, 2014).