Global solution for the coagulation equation of water droplets in atmosphere between two horizontal planes

TL;DR

This paper proves the global existence, uniqueness, and convergence to stationary state of solutions for a coagulation equation modeling water droplet density in the atmosphere, considering boundary conditions related to rainfall.

Contribution

It establishes a global existence and uniqueness theorem for the coagulation equation with specific boundary conditions and shows convergence to stationary solutions.

Findings

Proved global existence and uniqueness of solutions.

Showed convergence of solutions to stationary states.

Handled boundary conditions modeling rainfall.

Abstract

In this paper we give a global existence and uniqueness theorem for an initial and boundary value problem (IBVP) relative to the coagulation equation of water droplets and we show the convergence of the global solution to the stationary solution. The coagulation equation is an integro-differential equation that describes the variation of the density $\sigma$ of water droplets in the atmosphere. Furthermore, IBVP is considered on a strip limited by two horizontal planes and its boundary condition is such that rain fall from the strip. To obtain this result of global existence of the solution $\sigma$ in the space of bounded continuous functions, through the method of characteristics, we assume bounded continuous and small data, whereas the vector field, besides being bounded continuous, has $W^{1,\infty}-$ regularity in space.