Smoluchowski Coagulation Equation with a Flux of Dust Particles

TL;DR

This paper constructs and analyzes time-dependent solutions to the Smoluchowski coagulation equation with a constant dust flux, demonstrating existence, bounds, and convergence properties for a broad class of kernels.

Contribution

It introduces a method to construct flux solutions for general non-gelling kernels and establishes their bounds and convergence behavior over time.

Findings

Flux solutions have linearly increasing mass over time.

Explicit bounds for flux solutions are derived, showing they are controlled by a known explicit solution.

In the constant kernel case, flux solutions converge to a steady state as time progresses.

Abstract

We construct a time-dependent solution to the Smoluchowski coagulation equation with a constant flux of dust particles entering through the boundary at zero. The dust is instantaneously converted into particles and flux solutions have linearly increasing mass. The construction is made for a general class of non-gelling coagulation kernels for which stationary solutions, so-called constant flux solutions, exist. The proof relies on several limiting procedures on a family of solutions of equations with sources supported on ever smaller sizes. In particular, uniform estimates on the fluxes of these solutions are derived in order to control the singularity produced by the flux at zero. We further show that, up to the multiplication by a scalar, flux solutions averaged in size and integrated in time, are bounded from above by the explicit solution, $x^{-\frac{\gamma+3}{2}}$, of the constant flux equation, where $\gamma$ is the homogeneity of the bounds of the coagulation rate kernel. Flux solutions are expected to converge to a constant flux solution in the large time limit. We show that this is indeed true in the particular case of the constant kernel with zero initial data.