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Upper bounds on the coarsening rate of discrete, ill‐posed nonlinear diffusion equations
Author(s) -
Esedoḡlu Selim,
Greer John B.
Publication year - 2009
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20259
Subject(s) - mathematics , nonlinear system , diffusion , upper and lower bounds , space (punctuation) , population , statistical physics , mathematical analysis , flow (mathematics) , geometry , computer science , physics , demography , quantum mechanics , sociology , thermodynamics , operating system
We prove a weak upper bound on the coarsening rate of the discrete‐in‐space version of an ill‐posed, nonlinear diffusion equation. The continuum version of the equation violates parabolicity and lacks a complete well‐posedness theory. In particular, numerical simulations indicate very sensitive dependence on initial data. Nevertheless, models based on its discrete‐in‐space version, which we study, are widely used in a number of applications, including population dynamics (chemotactic movement of bacteria), granular flow (formation of shear bands), and computer vision (image denoising and segmentation). Our bounds have implications for all three applications. © 2008 Wiley Periodicals, Inc.