On a Nonlinear, Nonlocal Parabolic Problem with Conservation of Mass, Mean and Variance
On a Nonlinear, Nonlocal Parabolic Problem with Conservation of Mass, Mean and Variance
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DOI：
10.1080/03605302.2011.563402
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发表时间：
2011-08
影响因子：
1.9
通讯作者：
A. Tudorascu;Marcus Wunsch
中科院分区：
文献类型：
--
作者：
A. Tudorascu;Marcus Wunsch
在本文中，我们证明了某些多孔介质型泛函的最陡下降的二次Wasserstein距离的约束（但不是弱封闭）的流形上产生一个非线性，非局部抛物型偏微分方程连接到研究的渐近行为的过滤问题的解决方案。Carlen和Gangbo关于Wasserstein空间中负玻尔兹曼熵最速下降的约束优化的结果被推广到多孔介质型泛函。由此产生的福克-普朗克方程的一个有趣的特点是其漂移项的非局部性发生在同一时间，其非线性。
In this paper we prove that the steepest descent of certain porous-medium type functionals with respect to the quadratic Wasserstein distance over a constrained (but not weakly closed) manifold gives rise to a nonlinear, nonlocal parabolic partial differential equation connected to the study of the asymptotic behavior of solutions for filtration problems. The result by Carlen and Gangbo on constrained optimization for steepest descent of the negative Boltzmann entropy in the Wasserstein space is generalized to porous-medium type functionals. An interesting feature of the resulting Fokker-Planck equation is the nonlocality of its drift term occurring at the same time as its nonlinearity.