behavior of a fluid of electrons in a warm adiabatic fully ionized plasma when the magnetic field and the motion of the ions is neglected. Neglecting the magnetic field is customary and reduces the number of unknowns. Neglecting the ion motion is relevant since the ratio of the masses of the electrons and the ions is typically very small[^2]. We refer to [@Bit] for more on the physical background.

Equations represent a coupling of a compressible (inviscid) fluid with an electrostatic field. For the pure compressible Euler equation, even small and smooth initial perturbations of a constant equilibrium can lead to formation of shocks in finite time [@Sid].

However, here we show that the coupling with the self-consistent electric field stabilizes the system in the sense that small smooth perturbations of a constant background remain global and return to equilibrium. This phenomenon was first observed in $3D$ by Guo [@Guo]. We also refer to Guo-Pausader [@GuoPau] for a similar result for the ion equation and to the recent work of Germain-Masmoudi [@GerMas] for the complete Euler-Maxwell equation for the electrons in $3D$. On the other hand, large perturbations can lead to blow-up in finite time for the Euler-Poisson equation [@GuoTad].

