Abstract
We study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result shows that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term. In particular, we find that the diffusive kernel allows for the existence of stationary solutions while there cannot be any such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel power law lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.
Highlights
Atmospheric cluster formation processes [22], where certain species of the gas molecules can stick together and eventually produce macroscopic particles, are an important component in cloud formation and radiation scattering
The above cluster formation processes are modelled with the so-called General Dynamic Equation (GDE) [22]
We focus on the effect the addition of a source term has on solutions of standard one-component coagulation equations
Summary
Atmospheric cluster formation processes [22], where certain species of the gas molecules (called monomers) can stick together and eventually produce macroscopic particles, are an important component in cloud formation and radiation scattering. In the case of discrete problems, the distribution of clusters nα has been proved to behave in self-similar form for large times and for a large class of initial data if the kernel is constant, Kα,β = 1, or additive, Kα,β = α + β [32] For these kernels it is possible to find explicit representation formulas for the solutions of (1.2), (1.3) using Laplace transforms. In [10] it has been observed using a combination of asymptotic analysis arguments and numerical simulations that solutions of (1.2), (1.3) with a finite monomer density behave in self-similar form for long times and for a class of homogeneous coagulation kernels, even considering source terms which depend on time following a power law tω. We describe the asymptotics for large cluster sizes of these stationary solutions
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