TY - JOUR

T1 - On Global Stability of Optimal Rearrangement Maps

AU - Nguyen, Huy Q.

AU - Nguyen, Toan T.

N1 - Funding Information:
The authors thank Yann Brenier for introducing them the AHT model studied in this paper, Robert McCann for informing them the references [, ], and Dong Li for constructive comments. HN’s research was supported by NSF Grant DMS-1907776. TN’s research was supported in part by the NSF under Grant DMS-1764119 and by the 2018-2019 AMS Centennial Fellowship. Part of this work was done while TN was visiting the Department of Mathematics and the Program in Applied and Computational Mathematics at Princeton University.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/11/1

Y1 - 2020/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85087564942&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85087564942&partnerID=8YFLogxK

U2 - 10.1007/s00205-020-01552-0

DO - 10.1007/s00205-020-01552-0

M3 - Article

AN - SCOPUS:85087564942

VL - 238

SP - 671

EP - 704

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -