Instabilities and oscillations in coagulation equations with kernels of homogeneity one

TL;DR

This paper analyzes the long-time behavior of solutions to Smoluchowski's coagulation equation with homogeneity one kernels, revealing instabilities and oscillations depending on kernel structure, with implications for wave convergence.

Contribution

It provides a detailed analysis of the stability and oscillatory behavior of solutions for kernels near and far from the diagonal, connecting to Burgers equation dynamics.

Findings

Instability of constant solutions and traveling waves near diagonal kernels.

Strong oscillations in traveling waves for non-diagonal kernels.

Implications for the approach to N-waves with oscillatory behavior.

Abstract

We discuss the long-time behaviour of solutions to Smoluchowski's coagulation equation with kernels of homogeneity one, combining formal asymptotics, heuristic arguments based on linearization, and numerical simulations. The case of what we call diagonally dominant kernels is particularly interesting. Here one expects that the long-time behaviour is, after a suitable change of variables, the same as for the Burgers equation. However, for kernels that are close to the diagonal one we obtain instability of both, constant solutions and traveling waves and in general no convergence to N-waves for integrable data. On the other hand, for kernels not close to the diagonal one these structures are stable, but the traveling waves have strong oscillations. This has implications on the approach towards an N-wave for integrable data, which is also characterized by strong oscillations near the shock front.