Convergence and non-convergence phenomena in Euler-Maxwell to MHD transitions

In this work, we investigate the difference estimate for a class of Euler-Maxwell system and those of magnetohydrodynamics (in short, MHD) systems in three dimensions. We decompose the Euler-Maxwell system into three parts, namely the MHD system, auxiliary linear system and error part system. As a result, we obtain the convergence of the velocity of the fluid $u$, electric fields $E$ and magnetic fields $B$ from the Euler-Maxwell system toward the MHD system in $L^{p}_{t}L^{2}_{x}$ as the speed of light $c$ approaches infinity for $p\in[1,\infty]$. We also derived non-convergence results of electric current $j$ or $cE$, and these results are classified by a certain threshold for $p$. Finally, we investigate how the $L^2$-energy flow of Euler-Maxwell system evolves as c tends to infinity, leading to the vanishing of Amp\`ere's equation in the Euler-Maxwell system.