论文标题
在粗糙路径上流体动力学的变异原理
Variational principles for fluid dynamics on rough paths
论文作者
论文摘要
在本文中，我们在GFD中引入了一个新的参数化方案（PS）的框架。使用受控的粗糙路径理论，我们将一类粗糙的地球物理流体动力学（RGFD）模型作为粗糙作用功能的关键点。这些RGFD模型将流体动力学中的拉格朗日轨迹表征为差异图的流动性轨迹（GRP）。为这些模型的推导制定了三种约束的变分方法。首先是Clebsch公式，其中约束被施加为粗略的对流法。第二个是汉密尔顿 - 潘特拉金制剂，其中约束被施加为右转的粗糙矢量场。第三个是欧拉 - poincaré公式，其中变化受到限制。这些变分原理直接导致了lie-poisson hamiltonian对几何粗糙路径上流体动力学的表述。 GRP框架保留了通过使用谎言组还原从拉格朗日传递到欧拉的变异原理获得的流体动力学的几何结构，从而产生了开尔文循环定理的粗略表述。粗略的变化方法包括拉格朗日流体轨迹的非马克维亚扰动。特别是，可以通过这种表述来引入记忆效应，通过对粗糙路径的明智选择（例如，实现分数布朗运动）。在特殊情况下，当粗糙的路径是对半明星的实现时，我们在Holm（2015）中恢复了SGFD模型。但是，通过消除对随机变化工具的需求，我们保留了拉格朗日轨迹的路径解释。相反，随机框架中的拉格朗日轨迹由没有路径解释的随机积分来描述。因此，粗糙的路径公式恢复了此属性。
In this paper, we introduce a new framework for parametrization schemes (PS) in GFD. Using the theory of controlled rough paths, we derive a class of rough geophysical fluid dynamics (RGFD) models as critical points of rough action functionals. These RGFD models characterize Lagrangian trajectories in fluid dynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps. Three constrained variational approaches are formulated for the derivation of these models. The first is the Clebsch formulation, in which the constraints are imposed as rough advection laws. The second is the Hamilton-Pontryagin formulation, in which the constraints are imposed as right-invariant rough vector fields. The third is the Euler--Poincaré formulation in which the variations are constrained. These variational principles lead directly to the Lie--Poisson Hamiltonian formulation of fluid dynamics on geometric rough paths. The GRP framework preserves the geometric structure of fluid dynamics obtained by using Lie group reduction to pass from Lagrangian to Eulerian variational principles, thereby yielding a rough formulation of the Kelvin circulation theorem. The rough-path variational approach includes non-Markovian perturbations of the Lagrangian fluid trajectories. In particular, memory effects can be introduced through this formulation through a judicious choice of the rough path (e.g. a realization of a fractional Brownian motion). In the special case when the rough path is a realization of a semimartingale, we recover the SGFD models in Holm (2015). However, by eliminating the need for stochastic variational tools, we retain a pathwise interpretation of the Lagrangian trajectories. In contrast, the Lagrangian trajectories in the stochastic framework are described by stochastic integrals which do not have a pathwise interpretation. Thus, the rough path formulation restores this property.
