间接趋化级联中的结构二分性与质量临界性：四阶椭圆性与Volterra记忆
Structural dichotomy and mass criticality in indirect chemotaxis cascades: fourth-order ellipticity versus Volterra memory
AI总结：
本文通过分析Keller-Segel型趋化模型中的多阶段信号级联，揭示了瞬时平衡与瞬态动力学记忆之间的严格数学二分性，并证明了质量临界维度从N=2移至N=4，同时建立了相应的临界质量公式和空间-时间阈值理论。
摘要：
我们研究了Keller-Segel型趋化模型中间接信号生成机制的结构和算子理论基础。通过分析一个物理动机的多阶段信号级联，我们建立了瞬时平衡与瞬态动力学记忆之间的精确数学二分性。具体来说，我们证明了完全平衡的抛物-椭圆-椭圆（PES）级联简化为一个静态的四阶椭圆相互作用。在四维中，主导牛顿奇异性的精确代数抵消产生了一个纯对数核，将质量临界维度从$N=2$移至$N=4$。通过$L^2$-梯度流公式，我们确定了相应的浓度缩放候选临界质量$M_* = 64π^2τ/χ$。与此形成鲜明对比的是，我们证明了混合椭圆-抛物（MEP）级联保留了一个真正的Volterra记忆效应，该效应无法静态简化。其相互作用漂移在时间上表现为一个奇异扰动——在时间对角线附近表现出经典的二维Keller-Segel主阶，但在其冻结时间平均中提供四阶平滑，这需要一个混合的空间-时间阈值理论。这些结果分离了多尺度生物系统中质量临界维度移动的物理起源，并制定了封闭阈值问题所需的特定适应Adams/对数Hardy-Littlewood-Sobolev（log-HLS）不等式和混合范数准则。
英文摘要：
We investigate the structural and operator-theoretic foundations of indirect signal-generation mechanisms in Keller--Segel-type chemotaxis models. By analyzing a physically motivated multi-stage signaling cascade, we establish a precise mathematical dichotomy between instantaneous equilibration and transient kinetic memory. Specifically, we prove that the fully equilibrating parabolic--elliptic--elliptic (PES) cascade reduces to a static fourth-order elliptic interaction. In dimension four, an exact algebraic cancellation of the leading Newtonian singularity yields a purely logarithmic kernel, shifting the mass-critical dimension from $N=2$ to $N=4$. Through an $L^2$-gradient flow formulation, we identify the corresponding concentration-scaling candidate critical mass $M_* = 64π^2τ/χ$. In sharp contrast, we demonstrate that the mixed elliptic--parabolic (MEP) cascade retains a genuine Volterra memory effect that defies static reduction. Its interaction drift acts as a singular perturbation in time-exhibiting classical two-dimensional Keller--Segel principal order near the time diagonal, yet providing fourth-order smoothing in its frozen-time average, necessitating a mixed space-time threshold theory. These results isolate the physical origin of mass-critical dimensional shifts in multiscale biological systems and formulate the specific adapted Adams/logarithmic Hardy--Littlewood--Sobolev (log-HLS) inequalities and mixed-norm criteria required to close the threshold problems.
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