On Global Stability of Optimal Rearrangement Maps

Huy Q. Nguyen, Toan T. Nguyen

Research output: Contribution to journalArticle

Abstract

We study the nonlocal vectorial transport equation ∂ty+ (Py· ∇) y= 0 on bounded domains of Rd where P denotes the Leray projector. This equation was introduced to obtain the unique optimal rearrangement of a given map y as the infinite time limit of the solution with initial data y (Angenent et al.: SIAM J Math Anal 35:61–97, 2003; McCann: A convexity theory for interacting gases and equilibrium crystals. Thesis (Ph.D.)-Princeton University, ProQuest LLC, Ann Arbor, MI, p 163, 1994; Brenier: J Nonlinear Sci 19(5):547–570, 2009). We rigorously justify this expectation by proving that for initial maps y sufficiently close to maps with strictly convex potential, the solutions y are global in time and converge exponentially quickly to the optimal rearrangement of y as time tends to infinity.

Original languageEnglish (US)
Pages (from-to)671-704
Number of pages34
JournalArchive for Rational Mechanics and Analysis
Volume238
Issue number2
DOIs
StateAccepted/In press - 2020

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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