广义Fisher信息泛函关于修正Wasserstein距离的梯度流
The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances
摘要：
研究一类特殊的四阶偏微分方程非负弱解的存在性：该方程是关于广义Wasserstein迁移距离的形式梯度流，具有非线性迁移率.相应的自由能泛函被称为广义Fisher信息泛函，因为它是通过另一个能量泛函的自耗散而获得的，该能量泛函产生热流作为其相对于上述距离的梯度流。我们的主要结果有两个方面：对于满足一定正则性条件的迁移率函数，我们通过构造梯度流的极小化迁移方案，证明了弱解的存在性。此外，我们将这些结果推广到更一般的迁移率函数类：弱解可以通过正则化迁移率问题的弱解近似得到。
英文摘要：
This article is concerned with the existence of nonnegative weak solutions to a particular fourth-order partial differential equation: it is a formal gradient flow with respect to a generalized Wasserstein transportation distance with nonlinear mobility. The corresponding free energy functional is referred to as generalized Fisher information functional since it is obtained by autodissipation of another energy functional which generates the heat flow as its gradient flow with respect to the aforementioned distance. Our main results are twofold: For mobility functions satisfying a certain regularity condition, we show the existence of weak solutions by construction with the well-known minimizing movement scheme for gradient flows. Furthermore, we extend these results to a more general class of mobility functions: a weak solution can be obtained by approximation with weak solutions of the problem with regularized mobility.
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