论文标题
二维完美流体的最大混合平衡
On maximally mixed equilibria of two-dimensional perfect fluids
论文作者
论文摘要
二维完美（不可压缩和无粘性）流体的涡度由其保持流动的面积传输。给定初始的涡度分布$ω_0$，预测可以持续存在的长时间行为是基本重要性的问题。在无限的时间限制中，可能会发生一些不可逆的混合$ω_0$。由于动能$ \ Mathsf {e} $都是保守的，因此并非所有混合状态都是相关的，并且自然而然地考虑具有$ \ Mathsf {e} _0 $的能量$ \ Mathsf {e} _0 $对应于$ω_0$。由$ \叠加{\ Mathcal {o} _ {ω_0}}}^*\ cap \ {{\ Mathsf E} = {\ Mathsf E} _ _0 _0 \} $表示所有可能的最终状态。 A. Shnirelman介绍了最大混合状态的概念（任何进一步的混合都一定会改变其能量），并证明它们是完美的流体平衡。我们通过表明任何严格凸casimir以$ \ OPERLINE {\ MATHCAL {\ MATHCAL {O} _ {ω_0}}^*\ CAP \ cap \ {{\ Mathsf E} = {\ Mathsf E} _0的属性以及班级的水平混合，以相同的方式提供了对这一理论的新观点的新观点。理论。因此，（弱）融合到平衡不能仅仅基于动能的涡度传输和保护。另一方面，在具有对称的域上（例如直频道或环），我们利用了$ \ overline {\ Mathcal {o} _ {ω_0}}}}}^*^*\ cap \ cap \ cap \ {{\ Mathsf E} = {\ Mathsf E} = {可以任意接近$ l^1 $涡旋中的任何剪切或径向流，但不会在很长的时间内弱地收敛到它们。
The vorticity of a two-dimensional perfect (incompressible and inviscid) fluid is transported by its area preserving flow. Given an initial vorticity distribution $ω_0$, predicting the long time behavior which can persist is an issue of fundamental importance. In the infinite time limit, some irreversible mixing of $ω_0$ can occur. Since kinetic energy $\mathsf{E}$ is conserved, not all the mixed states are relevant and it is natural to consider only the ones with energy $\mathsf{E}_0$ corresponding to $ω_0$. The set of said vorticity fields, denoted by $\overline{\mathcal{O}_{ω_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$, contains all the possible end states of the fluid motion. A. Shnirelman introduced the concept of maximally 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 in $\overline{\mathcal{O}_{ω_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ is 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 (e.g. straight channel or annulus), we exploit all the conserved quantities and the characterizations of $\overline{\mathcal{O}_{ω_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ to give examples of open sets of initial data which can be arbitrarily close to any shear or radial flow in $L^1$ of vorticity but do not weakly converge to them in the long time limit.
