MR2589658 Kumiko HATTORI and Tetsuya HATTORI Kumiko HATTORI and Tetsuya HATTORI Funkcialaj Ekvacioj. Serio Internacia 52 2009 301--319 http://fe.math.kobe-u.ac.jp/FE/FullPapers/52-3/52_301.pdf http://www.ams.org/mathscinet-getitem?mr=MR2589658 We consider a system of partial differential equations for the densities of components of one dimensional incompressible fluid mixture whose motion is driven by evaporation. We prove existence and give explicit form of unique global classical non-negative solution to initial value problem for the system. The solution is known to be an infinite particle limit of stochastic ranking processes, which is a simple stochastic model of time evolutions of Amazon Sales Ranks. As a practical application, we collected data from the web and performed statistical fits to our formula. The results suggest that the system of equations and solutions, though very simple, may have applications in the analysis of online rankings. Evaporation driven fluid, Non-linear wave, Stochastic ranking process, Long tail, Online ranking. Primary 35C05; Secondary 35Q35, 82C22. 52-301 2009 Kumiko HATTORI and Tetsuya HATTORI Kumiko HATTORI and Tetsuya HATTORI 1 Anderson, C. The Long Tail: Why the Future of Business Is Selling Less of More Hyperion Books 2006

2 Brynjolfsson, E.; Hu, Y.; Smith, M. D. Management Science 49 2003 1580--1596

3 Chevalier, J.; Goolsbee, A. Quantitative Marketing and Economics 1 2003 203--222

4 Hattori, K.; Hattori, T. Stochastic Process. Appl. 119 2009 966--979 MR2499866

5 Hattori, K.; Hattori, T., Mathematical analysis of long tail economy using stochastic ranking processes, preprint, http://arxiv.org/abs/0804.0321 6 Rosenthal, M., http://www.fonerbooks.com/surfing.htm 7 Rosenthal, M. Print-On-Demand Book Publishing: A New Approach to Printing and Marketing Books for Publishers and Self-Publishing Authors Foner Books 2004