论文标题
广义流体流的几何形状
Geometry of generalized fluid flows
论文作者
论文摘要
在V. Arnold之后，可以将理想（即无可挑剔的不可压缩）流体的Euler方程视为在流量域中的右界不变$ l^2 $ metric的测量流量。在本文中，我们描述了广义流，多相流体（均质涡流片）和常规涡流片的共同起源和对称性：它们都对应于多形差异性的某些组素质的大地测量。此外，我们证明，所有这些问题都是哈密顿式的，就双重谎言代数的泊松结构而言，在谎言代数二重奏上概括了欧拉方程的哈密顿属性。
The Euler equation of an ideal (i.e. inviscid incompressible) fluid can be regarded, following V.Arnold, as the geodesic flow of the right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. In this paper we describe the common origin and symmetry of generalized flows, multiphase fluids (homogenized vortex sheets), and conventional vortex sheets: they all correspond to geodesics on certain groupoids of multiphase diffeomorphisms. Furthermore, we prove that all these problems are Hamiltonian with respect to a Poisson structure on a dual Lie algebroid, generalizing the Hamiltonian property of the Euler equation on a Lie algebra dual.
