Long-term regularity of the periodic Euler--Poisson system for electrons in 2D

We study a basic plasma physics model–the one-fluid Euler–Poisson system on the square torus, in which a compressible electron fluid flows under its own electrostatic field. In this paper we prove long-term regularity of periodic solutions of this system in 2 spatial dimensions. Our main conclusion is that on a square torus of side length $R$, if the initial data is sufficiently close to a constant solution, then the solution is wellposed for a time at least $R\mathrm{/}{ϵ}^2(\log R{)}^{O(1)}$, where $ϵ$ is the size of the initial data.