Long-time behaviour for kinetic models of clustering and non-local diffusion equations
Final Report Abstract
This project was concerned with the investigation of the long-time behaviour in coalescing particle systems. Such processes can be observed in many situations in the natural sciences. Typical examples are the formation of rain drops in clouds and polymerisation in chemistry but also include growth processes for systems of microorganisms or large-scale effects in astrophysics such as the formation of planets. An important mathematical model describing these phenomena is given by Smoluchowski’s coagulation equation. It has been conjectured for a long time that solutions to Smoluchowski’s equation exhibit a self-similar form. However, except from few explicitly solvable cases, the conjecture could not be verified. In the course of this project several results concerning this problem have been obtained. Most important, we have been able to provide the first verification of the scaling hypothesis for a class of models which does not allow for explicit solution formulas. More precisely, for small variations of the solvable constant coagulation kernel, we proved that solutions approach a self-similar profile for large times. Even more, we gave an estimate on the speed at which this self-similar form is attained. A further result concerns the uniqueness of self-similar profiles. Again, we have been considering a perturbation of the constant coagulation kernel (which can be unbounded in this case) and we proved that there can only be one self-similar solution. This result improves an earlier work by allowing for a much less restrictive class of perturbations while, at the same time, the proof itself could be significantly simplified. In a third work, we reconsidered the solvable models and we provided a new contractivity estimate for the corresponding solutions in distances based on the Laplace transform.
Publications
- Power Network Dynamics on Graphons. SIAM J. Appl. Math., 79(4):1271–1292, 2019
C. Kuehn and S. Throm
(See online at https://doi.org/10.1137/18M1200002) - Smoluchowski’s discrete coagulation equation with forcing. NoDEA Nonlinear Differential Equations Appl., 26(3):26:17, 2019
C. Kuehn and S. Throm
(See online at https://doi.org/10.1007/s00030-019-0563-9) - Stability and uniqueness of self-similar profiles in L1 spaces for perturbations of the constant kernel in Smoluchowski’s coagulation equation. Communications in Mathematical Physics, 2019
S. Throm
- Contractivity for Smoluchowski’s coagulation equation with solvable kernels. Bulletin of the London Mathematical Society, 2020
J. A. Cañizo, B. Lods, and S. Throm
(See online at https://doi.org/10.1112/blms.12417) - The scaling hypothesis for Smoluchowski’s coagulation equation with bounded perturbations of the constant kernel. J. Differential Equations, 270:285–342, 2021
J. A. Cañizo and S. Throm
(See online at https://doi.org/10.1016/j.jde.2020.07.036)
