Density Collapse and Gradient Catastrophe in a One-Dimensional Euler--Poisson--Cattaneo System

Abstract

We study finite-time singularity formation for a one-dimensional Euler--Poisson system in Lagrangian coordinates with Cattaneo heat conduction and quadratic heat-flux corrections in the pressure and internal energy. Under the structural choice $κ=κ_0v$, two different breakdown mechanisms are obtained. First, an explicit affine solution reaches $v=0$ in finite time, so the Eulerian density $ρ=1/v$ diverges while the remaining state variables stay finite at each fixed spatial point. Second, for $a'(1)<0$, the Poisson field is localized near equilibrium by $F=ϕ_x/v$, producing a strictly hyperbolic balance law subject to the propagated constraint $F_x=1-v$. We compute the characteristic speeds and an explicit genuine-nonlinearity coefficient and show that genuine nonlinearity fails for all propagating families only when two explicit algebraic degeneracy conditions hold simultaneously. For neutral compactly supported data satisfying the Poisson constraint exactly, a narrow compressive wave generates only an $O(\varepsilon^2η)$ zero-speed Poisson mode. A constraint-compatible John--Hörmander--Bärlin bootstrap and a Riccati comparison then yield finite-time gradient catastrophe while the solution remains uniformly close to equilibrium and $F_x$ stays bounded. The two results exhibit density collapse and small-amplitude gradient blow-up as distinct breakdown mechanisms within the same Euler--Poisson--Cattaneo model.