A complete characterization of sharp thresholds to spherically symmetric multidimensional pressureless Euler-Poisson systems

The Euler-Poisson (EP) system models the dynamics of a variety of physical processes, including charge transport, collisional plasmas, and certain cosmological wave phenomena. In this work, we establish sharp critical threshold conditions that distinguish global-in-time regularity from finite-time breakdown for solutions of the radially symmetric, multidimensional pressureless EP system. Overall, there are two cases: with and without background ($c>0, c=0$ respectively). For $c>0$, we obtain precise thresholds assuming a periodicity condition. A key feature of our approach is that it extends seamlessly to the zero background case, where we obtain sharp thresholds without imposing any additional assumptions. In particular, the framework accommodates initial velocities that may be negative, allowing the flow to be directed toward the origin. The main analytical challenge of deriving threshold conditions for EP systems stems from the intricate coupling of various local/nonlocal forces. To overcome this, we identify a novel nonlinear quantity that plays a decisive role in the analysis and enables a unified treatment of all relevant scenarios. Our results provide a comprehensive characterization of critical thresholds for the pressureless EP system in multiple dimensions.

💡 Research Summary

The paper presents a comprehensive analysis of critical thresholds for the pressureless Euler‑Poisson (EP) system under spherical symmetry in multiple spatial dimensions. The authors consider both the presence (c > 0) and absence (c = 0) of a uniform background charge and derive sharp, explicit conditions that separate global‑in‑time regularity from finite‑time breakdown (shock or vacuum formation).

Starting from the full EP system (continuity, momentum, and Poisson equations) they impose radial symmetry, reducing the PDEs to a one‑dimensional system in the radial variable r. Introducing the variables ρ (density), p = u_r (radial velocity gradient), q = u_r (the same notation is used for convenience), and s = −ϕ_r, the dynamics along each characteristic curve can be written as a coupled 4‑dimensional ODE system. The q–s subsystem decouples and admits a conserved quantity R_N(q, ŝ) (with ŝ = s + c/N), which guarantees that (q, s) evolve on closed, periodic trajectories provided the initial s₀ > −c/N. This periodicity yields the auxiliary factor

\