On Global Stability of Optimal Rearrangement Maps

We study the nonlocal vectorial transport equation ∂_ty+ (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.