Is gelation a singularity or a flow induced instability?

Manuel Dedola; Ludovico Cademartiri

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arXiv · 2601.18806

Is gelation a singularity or a flow induced instability?

Manuel Dedola·Ludovico Cademartiri

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Abstract

Gelation in the Smoluchowski coagulation equation is commonly interpreted as a finite-time singularity marked by mass loss or moment divergence. We instead characterize gelation as a loss of dynamical stability of the Smoluchowski flow, quantified through the time-dependent spectrum of the Jacobian along the evolving aggregation dynamics. Studying homogeneous kernels $K(i,j)=(ij)^{\alpha}$ together with the classical Smoluchowski, we show that gelation is consistently preceded by the appearance of positive real eigenvalues, indicating a loss of local dynamical stability. While non-gelling kernels exhibit only transient finite-size effects, gelling kernels display persistent spectral destabilization associated with macroscopic gel formation. Our results identify gelation as a genuine dynamical instability of the Smoluchowski flow.

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Manuel Dedola, Ludovico Cademartiri. 2026-01-15. Is gelation a singularity or a flow induced instability?. https://arxiv.org/abs/2601.18806

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