### Abstract

We study the nonlocal vectorial transport equation ∂_{t}y+ (Py· ∇) y= 0 on bounded domains of R^{d} 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 language | English (US) |
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Pages (from-to) | 671-704 |

Number of pages | 34 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 238 |

Issue number | 2 |

DOIs | |

State | Accepted/In press - 2020 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering

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## Cite this

*Archive for Rational Mechanics and Analysis*,

*238*(2), 671-704. https://doi.org/10.1007/s00205-020-01552-0