Equation of motion for incompressible mixed fluid driven by evaporation and its application to online rankings

We give a unique classical solution to initial value problem for a system of partial differential equations for the densities of components of one dimensional incompressible fluid mixture driven by evaporation. Motivated by the known fact that the solution appears as an infinite particle limit of stochastic ranking processes, which is a simple stochastic model of time evolutions of e.g., Amazon Sales Ranks, we collected data from the web and performed statistical fits to our formula. The results suggest that the fluid equations and solutions may have an application in the analysis of online rankings.