On Maximally Mixed Equilibria of Two-Dimensional Perfect Fluids
On Maximally Mixed Equilibria of Two-Dimensional Perfect Fluids
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DOI：
10.1007/s00205-022-01825-w
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发表时间：
2022-04
影响因子：
2.5
通讯作者：
Michele Dolce;Theodore D. Drivas
中科院分区：
文献类型：
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作者：
Michele Dolce;Theodore D. Drivas
二维理想(不可压缩无粘)流体的涡量是通过其保面积流动来输运的。在给定初始涡度分布的情况下，预测能够持续的长时间行为是一个基本的重要问题。在无限的时间内，可能会发生一些不可逆的混合。由于动能守恒，并不是所有的混合态都是相关的，所以只考虑能量对应的混合态是很自然的。由表示的所述涡量场的集合包含流体运动的所有可能的结束状态。A.Shnirelman引入了最大混合态的概念(任何进一步的混合必然会改变它们的能量)，并证明了它们是完美的流体平衡。我们证明了任何严格凸Casimir空间的极小元是极大混合的，并讨论了它与经典统计流体力学理论的关系，从而为这一理论提供了一个新的视角。因此，不能仅仅因为涡度输送和动能守恒而排除(弱)收敛到平衡。另一方面，在具有对称性的区域(例如直槽或环空)上，我们利用所有守恒量和特征，给出了开放的初始数据集的例子，它们可以在涡度范围内任意接近任何切变或径向流动，但在长时间限制下不弱收敛于它们。
The vorticity of a two-dimensional perfect (incompressible and inviscid) fluid is transported by its area preserving flow. Given an initial vorticity distribution, predicting the long time behavior which can persist is an issue of fundamental importance. In the infinite time limit, some irreversible mixing ofcan occur. Since kinetic energyis conserved, not all the mixed states are relevant and it is natural to consider only the ones with energycorresponding to. The set of said vorticity fields, denoted by, contains all the possible end states of the fluid motion. A. Shnirelman introduced the concept ofmaximally mixed states(any further mixing would necessarily change their energy), and proved they are perfect fluid equilibria. We offer a new perspective on this theory by showing that any minimizer of any strictly convex Casimir inis maximally mixed, as well as discuss its relation to classical statistical hydrodynamics theories. Thus, (weak) convergence to equilibrium cannot be excluded solely on the grounds of vorticity transport and conservation of kinetic energy. On the other hand, on domains with symmetry (for example straight channel or annulus), we exploit all the conserved quantities and the characterizations ofto give examples of open sets of initial data which can be arbitrarily close to any shear or radial flow inof vorticity but do not weakly converge to them in the long time limit.