The Spectrum of a Family of Fourth-Order Nonlinear Diffusions Near the Global Attractor
The Spectrum of a Family of Fourth-Order Nonlinear Diffusions Near the Global Attractor
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全局吸引子附近四阶非线性扩散族的谱
DOI：
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发表时间：
2015
期刊：
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通讯作者：
Christian Seis
中科院分区：
文献类型：
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作者：
R. McCann;Christian Seis
薄膜和量子漂移扩散方程属于[21]中提出的四阶演化方程族，类似于（二阶）多孔介质族。它们是广义Fisher信息I（v）的2-Wasserstein（= d2）梯度流，正如Otto [41]证明多孔介质族是广义熵E（v）的d2梯度流一样。恒等式<$E（v）= bE（v）+|D 2 E（v）|2/2形式上意味着当方程围绕所得的自相似临界轮廓v * 重新标度和线性化时，a Hess d 2 I（v *）= Hess d 2 E（v *）（B + Hess d 2 E（v *））。我们将这个关系与[46]中计算的多孔介质流动的Hess d 2 E（v *）的对角化相结合。这将产生有关的领导和高阶渐近方程的R N的特殊情况之外，以前是无法访问的信息。
The thin film and quantum drift diffusion equations belong to a fourth-order family of evolution equations proposed in [21] to be analogous to the (second-order) porous medium family. They are 2-Wasserstein (=d 2) gradient flows of the generalized Fisher information I(v) just as the porous medium family was shown to be the d 2 gradient flow of the generalized entropy E(v) by Otto [41]. The identity aI(v) = bE(v) + |∇ d 2 E(v)|2/2 implies a Hess d 2 I(v *) = Hess d 2 E(v *)(b + Hess d 2 E(v *)) formally, when the equation is rescaled and linearized around the resulting self-similar critical profile v *. We couple this relation with the diagonalization of Hess d 2 E(v *) for the porous medium flow computed in [46]. This yields information about the leading- and higher-order asymptotics of the equation on R N which—outside of special cases—was inaccessible previously.