论文标题
欧拉的流体方程中有限时间奇异性的哈密顿量描述
A Hamiltonian description of finite-time singularity in Euler's fluid equations
论文作者
论文摘要
最近提出的Moffat和Kimura [1,2]的低自由度模型用于描述不可压缩的Euler流体方程的有限时间奇异性的方法。该模型假设两个涡流环在两个倾斜平面上对称放置的两个涡旋环的初始有限能量构型。获得了模型的无粘性极限的哈密顿结构。发现了相关的非规范泊松托架[3]和两个不变的人，一个是汉密尔顿人，另一个是卡西米尔不变的。结果表明，该系统与位于两个不变的交点上的解决方案可以集成，就像将解决方案的自由刚性机理相同，其解决方案位于动能和角动量表面的交汇处。同样，给出了直接的正交，并用于演示模型中有限时间奇异性的leray形式。在一定程度上，Moffat和Kimura模型准确地代表了Euler的理想流体运动方程，我们已经表明存在有限的奇异性。
The recently proposed low degree-of-freedom model of Moffat and Kimura [1,2] for describing the approach to finite-time singularity of the incompressible Euler fluid equations is investigated. The model assumes an initial finite-energy configuration of two vortex rings placed symmetrically on two tilted planes. The Hamiltonian structure of the inviscid limit of the model is obtained. The associated noncanonical Poisson bracket [3] and two invariants, one that serves as the Hamiltonian and the other a Casimir invariant, are discovered. It is shown that the system is integrable with a solution that lies on the intersection for the two invariants, just as for the free rigid body of mechanics whose solution lies on the intersection of the kinetic energy and angular momentum surfaces. Also, a direct quadrature is given and used to demonstrate the Leray form for finite-time singularity in the model. To the extent the Moffat and Kimura model accurately represents Euler's ideal fluid equations of motion, we have shown the existence of finite-time singularity.
