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

Abstract

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 $$(0,\vec {e}_1)$$ , $$\vec {e}_1=(1,0)$$ ). Precisely, they showed some explicit decay rates of solutions in $$L^2$$ norm. So a nature question is whether the obtained decay rates in 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 $$L^\infty $$ 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).