Asymptotic stability of explicite infinite energy blowup solutions for three dimensional incompressible Magnetohydrodynamics equations

Abstract

This paper is denoted to the study of dynamical behavior near explicit finite time blowup solutions for three dimensional incompressible Magnetohydrodynamics (MHD) equations. More precisely, we find a family of explicit finite time blowup solutions admitted smooth initial data and infinite energy in whole space $\mathbb{R}^3$. After that, we prove asymptotic stability of those explicit finite time blowup solutions for $3$D incompressible Magnetohydrodynamics equations in a smooth bounded domain with free surface $$ \Omega_{t}:=\Big\{(t,x_1,x_2,x_3):0\leq x_i\leq\sqrt{\overline{T}^*-t},\quad t\in(0,\overline{T}^*),\quad i=1,2,3\Big\}, $$ where $\overline{T}^*$ denotes the blowup time. This means we construct a family of \textbf{stable} blowup solutions for $3$D incompressible Magnetohydrodynamics equations with smooth initial data in $\Omega_t$.