发表机构
Nanyang Institute of Technology（南阳理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分析一维Euler--Poisson--Cattaneo系统的有限时间奇点，发现密度坍缩与梯度灾变两种不同破裂机制，并给出严格数学证明。
AI 中文摘要
我们研究了拉格朗日坐标下具有Cattaneo热传导以及压力和内能中二次热流修正的一维Euler--Poisson系统的有限时间奇点形成。在结构选择$\kappa=\kappa_0v$下，获得了两种不同的破裂机制。首先，一个显式仿射解在有限时间内达到$v=0$，因此欧拉密度$\rho=1/v$发散，而其余状态变量在每个固定空间点保持有限。其次，对于$a'(1)<0$，泊松场通过$F=\phi_x/v$在平衡态附近局部化，产生一个严格双曲平衡律，并受传播约束$F_x=1-v$支配。我们计算了特征速度和显式的真正非线性系数，并表明仅当两个显式代数退化条件同时成立时，所有传播族的真正非线性才失效。对于满足泊松约束精确成立的紧支撑中性数据，一个窄压缩波仅产生$O(\varepsilon^2\eta)$的零速泊松模式。一个约束相容的John--Hörmander--Bärlin自举和Riccati比较随后产生有限时间梯度灾变，而解保持均匀接近平衡态且$F_x$保持有界。这两个结果展示了在同一Euler--Poisson--Cattaneo模型中，密度坍缩和小振幅梯度爆破作为不同的破裂机制。
英文摘要
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.
