| Abstract: |
| The Smoluchowski coagulation equation is a classical equation
describing the distribution of particle sizes undergoing binary
coagulation. It arises in many areas of science, including aerosol
science, molecular biology, and ecology. Despite extensive study for
over a century, it continues to pose significant analytical
challenges, particularly concerning long-time behavior for general
coagulation kernels.
Recently, an existence theory for nontrivial steady states has been
developed for a large class of kernels describing coagulation in open
systems. We construct a time-dependent solution that is expected to
converge to a nontrivial steady state, and prove this convergence for
the constant kernel with zero initial data. The solution satisfies a
nonzero flux boundary condition describing a constant input of dust
particles. We show that this dust is instantaneously converted into
particles, so that none remains in the system, and the total mass
grows linearly in time. The construction applies to a broad class of
non-gelling kernels for which stationary solutions exist; in the
complementary regime, no such type of solutions with flux exist.
(Based on a joint work with Aleksis Vuoksenmaa - Sapienza University) |
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