First, we prove uniform-in- $ϵ$ regularity estimates of local strong solutions to the Chern-Simons-Schrödinger equations in ${\mathbb{R}}^2$. Here $ϵ$ is the dispersion coefficient. Then we prove the global well-posedness of strong solutions to the limit problem $(ϵ=0)$.

In this paper, we will use the Banach fixed point theorem to prove the uniform-in- $ϵ$ existence of the compressible full magneto-micropolar system in a bounded smooth domain, where $ϵ$ is the dielectric constant. Consequently, the limit as $ϵ\to 0$ can be established. This approximation is usually referred to as the magnetohydrodynamics approximation and is equivalent to the neglect of the displacement current.

This paper poses a new question and proves a related result. Particularly, the nonexistence of a nontrivial time-periodic solution to the Navier–Stokes system is proved in a bounded domain in ${\mathbb{R}}^2$.