$n\geq 2$维理想不可压缩流体模型的弱对偶对和Jetlet方法
Weak dual pairs and jetlet methods for ideal incompressible fluid models in $n\geq 2$ dimensions
摘要：
我们回顾了对偶对在力学中的作用，并利用它们来推导正则化不可压缩流体系统的类粒子解。在我们的例子中，我们有一个对偶对，它是由双同态作用于点粒子（实质上是通过移动点）而产生的。然后，我们通过考虑泰勒级数上的非同态的作用来增加我们的对偶对，也称为喷流。增广的弱对偶诱导的粒子类的解决方案和守恒律与粒子携带一个副本的喷流组的层次。我们称这些增广粒子为小喷流。喷流群作为有限维模型的自同构群本身，因此喷流粒子作为有限维模型的自相似性所表现出的理想不可压缩流体。的守恒定律相关的射流解决方案是一个阴影的开尔文环流定理。最后，我们研究了无限时间粒子并合的动力学。我们证明了两个合并的粒子在第零级的层次产生的动力学渐近接近的一个单一的粒子在第一级的层次。这种合并行为，然后验证数值以及角动量的交换，必须发生在两个粒子的近碰撞。由此产生的粒子样的解决方案提出了一类新的无网格方法，工作在尺寸$n \geq 2$，并表现出开尔文的环流定理的阴影。更广泛地说，这提供了理想流体中自相似性的第一个有限维模型之一。
英文摘要：
We review the role of dual pairs in mechanics and use them to derive particle-like solutions to regularized incompressible fluid systems. In our case we have a dual pair resulting from the action of diffeomorphisms on point particles (essentially by moving the points). We then augment our dual pair by considering the action of diffeomorphisms on Taylor series, also known as jets. The augmented weak dual pairs induce a hierarchy of particle-like solutions and conservation laws with particles carrying a copy of a jet group. We call these augmented particles jetlets. The jet groups serve as finite-dimensional models of the diffeomorphism group itself, and so the jetlet particles serve as a finite-dimensional model of the self-similarity exhibited by ideal incompressible fluids. The conservation law associated to jetlet solutions is shown to be a shadow of Kelvin's circulation theorem. Finally, we study the dynamics of infinite time particle mergers. We prove that two merging particles at the zeroth level in the hierarchy yield dynamics which asymptotically approach that of a single particle in the first level in the hierarchy. This merging behavior is then verified numerically as well as the exchange of angular momentum which must occur during a near collision of two particles. The resulting particle-like solutions suggest a new class of meshless methods which work in dimensions $n \geq 2$ and which exhibit a shadow of Kelvin's circulation theorem. More broadly, this provides one of the first finite-dimensional models of self-similarity in ideal fluids.
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