The Landau equation as a gradient Flow
The Landau equation as a gradient Flow
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作为梯度流的朗道方程
DOI：
10.2140/apde.2024.17.1331
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Jeremy Wu
中科院分区：
文献类型：
--
作者：
J. Carrillo;M. Delgadino;L. Desvillettes;Jeremy Wu
我们提出了软势空间齐次朗道方程的梯度流视角。我们基于朗道方程的熵耗散在概率测度空间上构建了一个定制的度量。在此度量下，朗道方程可以表征为玻尔兹曼熵的梯度流。特别是，我们通过函数不等式（通常称为能量耗散不等式）来表征偏微分方程的动力学。此外，与最优交通设置类似，我们表明这种解释可以用于最小化运动方案，以构造正则化朗道方程的解。
We propose a gradient flow perspective to the spatially homogeneous Landau equation for soft potentials. We construct a tailored metric on the space of probability measures based on the entropy dissipation of the Landau equation. Under this metric, the Landau equation can be characterized as the gradient flow of the Boltzmann entropy. In particular, we characterize the dynamics of the PDE through a functional inequality which is usually referred as the Energy Dissipation Inequality. Furthermore, analogous to the optimal transportation setting, we show that this interpretation can be used in a minimizing movement scheme to construct solutions to a regularized Landau equation.