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On Global Stability of Optimal Rearrangement Maps
Archive for Rational Mechanics and Analysis ( IF 2.5 ) Pub Date : 2020-07-06 , DOI: 10.1007/s00205-020-01552-0
Huy Q. Nguyen , Toan T. Nguyen

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 the initial map $y_0$ as its steady states (\cite{AHT, Macthesis, Brenier09}). 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 fast to the optimal rearrangement of $y_0$ as time tends to infinity.

中文翻译:

关于最优重排图的全局稳定性

我们在 $\mathbb{R}^d$ 的有界域上研究非局部矢量传输方程 $\partial_ty+ (\mathbb{P} y \cdot \nabla) y=0$ 其中 $\mathbb{P}$ 表示 Leray投影仪。引入这个方程是为了获得初始映射 $y_0$ 作为其稳态的唯一最优重排(\cite{AHT, Macthesis, Brenier09})。我们通过证明对于初始地图​​ $y_0$ 足够接近具有严格凸势的地图,解决方案 $y$ 在时间上是全局的,并且随着时间趋于无穷大,以指数方式快速收敛到 $y_0$ 的最佳重排,从而严格证明了这一期望。
更新日期:2020-07-06
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