Existence, Uniqueness and Regularity of a Time-Periodic Probability Density Distribution Arising in a Sedimentation-Diffusion Problem

Abstract

An analysis is presented of the one-dimensional convective-diffusive equation governing the temporal evolution and spatial distribution of the probability density $P(x,t)$ describing the sedimentation and diffusion of a nonneutrally buoyant Brownian particle in a vertical fluid-filled cylinder that is flipped over instantaneously at regular intervals. A time-periodic solution is sought by requiring that the initial spatial distribution recur after one complete period of the flipping motion. The periodicity condition is formulated both as an infinite matrix equation for the eigenfunction expansion coefficients of a possible recurring initial distribution, and as an integral equation of the second kind for the distribution itself. The kernel (infinite matrix) is shown to be square integrable (square summable), so that Fredholm theory applies. There exists a unique time-periodic solution whenever unity is not an eigenvalue of the integral (infinite matrix) operator. Regions in parameter space are identified ...