A semigroup approach to an integro-differential equation modeling slow erosion

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Abstract

The paper is concerned with a scalar conservation law with nonlocal flux, providing a model for granular flow with slow erosion and deposition. While the solution u= u(t, x) can have jumps, the inverse function x= x(t, u) is always Lipschitz continuous; its derivative has bounded variation and satisfies a balance law with measure-valued sources. Using a backward Euler approximation scheme combined with a nonlinear projection operator, we construct a continuous semigroup whose trajectories are the unique entropy weak solutions to this balance law. Going back to the original variables, this yields the global well-posedness of the Cauchy problem for the granular flow model.

Original languageEnglish (US)
Pages (from-to)2360-2403
Number of pages44
JournalJournal of Differential Equations
Volume257
Issue number7
DOIs
StatePublished - Oct 1 2014

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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