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On Global Stability of Optimal Rearrangement Maps

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Abstract

We study the nonlocal vectorial transport equation \(\partial _ty+ ({\mathbb {P}}y \cdot \nabla ) y=0\) on bounded domains of \({\mathbb {R}}^d\) where \({\mathbb {P}}\) denotes the Leray projector. This equation was introduced to obtain the unique optimal rearrangement of a given map \(y_0\) as the infinite time limit of the solution with initial data \(y_0\) (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 \(y_0\) sufficiently close to maps with strictly convex potential, the solutions y are global in time and converge exponentially quickly to the optimal rearrangement of \(y_0\) as time tends to infinity.

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Acknowledgements

The authors thank Yann Brenier for introducing them the AHT model studied in this paper, Robert McCann for informing them the references [12, 13], 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.

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Authors and Affiliations

  1. Department of Mathematics, Brown University, Providence, RI, 02912, USA

    Huy Q. Nguyen

  2. Department of Mathematics, Penn State University, State College, PA, 16803, USA

    Toan T. Nguyen

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  1. Huy Q. Nguyen
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  2. Toan T. Nguyen
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Correspondence to Huy Q. Nguyen.

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Communicated by P. Constantin

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Nguyen, H.Q., Nguyen, T.T. On Global Stability of Optimal Rearrangement Maps. Arch Rational Mech Anal 238, 671–704 (2020). https://doi.org/10.1007/s00205-020-01552-0

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