Upper Bounds on Coarsening Rates in Demixing Binary Viscous Liquids
Upper Bounds on Coarsening Rates in Demixing Binary Viscous Liquids
复制标题
二元粘性液体分层中粗化速率的上限
DOI：
10.1137/090775142
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发表时间：
2011
期刊：
影响因子：
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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：
Y. Brenier;F. Otto;Christian Seis
我们考虑了二元粘性液体在温度淬火后的粗化过程。在第一种以扩散为主的粗化过程(“蒸发-再冷凝过程”)中，典型的长度尺度按幂定律增加，其中t是时间。西吉亚[Phys.Rev.A，20(1979)，pp.595-605]认为在第二种状态下，粗化应该通过混合物的粘性流动来调节。这导致粗化率与幂定律$\ell\sim t$交叉。我们考虑了一个简单的尖锐界面模型，该模型只考虑了流动中介的粗化。对于这个模型，我们严格地证明了粗化不能比$\ell\sim t$进行得更快。这一分析与[R.V.Kohn和F.Otto，Comm.数学课。Phys，229(2002)，pp.375-395]，这是基于梯度流结构的演化。该分析利用具有对数成本函数的Monge-Kantorowicz-Rubinstein输运距离作为本征输运距离的代理.
We consider the coarsening process of a binary viscous liquid after a temperature quench. In a first, diffusion-dominated coarsening regime (“evaporation-recondensation process”), the typical length scale $\ell$ increases according to the power law $\ell\sim t^{1/3}$, where t is the time. Siggia [Phys. Rev. A, 20 (1979), pp. 595–605] argued that in a second regime, coarsening should be mediated by viscous flow of the mixture. This leads to a crossover in the coarsening rates to the power law $\ell\sim t$. We consider a simple sharp-interface model which just allows for flow-mediated coarsening. For this model, we prove rigorously that coarsening cannot proceed faster than $\ell\sim t$. The analysis follows closely a method proposed in [R. V. Kohn and F. Otto, Comm. Math. Phys., 229 (2002), pp. 375–395], which is based on the gradient flow structure of the evolution. The analysis makes use of a Monge–Kantorowicz–Rubinstein transportation distance with logarithmic cost function as a proxy for the intrinsic ...