A note on the non-existence of small non-trivial compact solutions for Euler-Poisson equation in 1D

In this short note, we prove the non-existence of slow and fast small nontrivial compact solutions for the Euler-Poisson system in $1$D. The proof is based on the virial estimate which provides local in space average decay of bounded small solutions.

💡 Research Summary

In this short note the authors address the question of whether non‑trivial compactly supported (or “compact”) solutions can exist for the one‑dimensional Euler–Poisson system describing ion dynamics. After a standard reformulation of the system in terms of the density perturbation (n=\rho-1), the velocity (u) and the electric potential (\phi), they introduce an energy density \