Nonlocal Dynamics of Passive Tracer Dispersion with Random Stopping
Jinqiao Duan, James R. Brannan, H. Gao
TL;DR
The paper addresses how random trapping sites with memory effects alter passive tracer dispersion in fluids, leading to a nonlocal advection-diffusion equation with a memory term. It develops an energy-method framework to bound the concentration in $L^2$ and shows that, under conditions on the velocity field $U(x)$ and the memory kernel $k(t)$ (notably bounded $U'$ and strictly increasing $k$), the mean-square concentration decays exponentially. The work also extends the analysis to a history-source scenario by reformulating the system into an autonomous, higher-dimensional state $(C, \eta^t)$ and proving exponential decay for the combined state. These results provide rigorous decay rates for pollutant dispersion in environments with traps and memory, with implications for coastal and riverine environmental modeling and memory-term transport analyses.
Abstract
We investigate the nonlocal behavior of passive tracer dispersion with random stopping at various sites in fluids. This kind of dispersion processes is modeled by an integral partial differential equation, i.e., an advection-diffusion equation with a memory term. We have shown the exponential decay of the passive tracer concentration, under suitable conditions for the velocity field and the probability distribution of random stopping time.