发表机构
University of Saskatchewan（萨斯喀彻温大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完整分类了三维不可压缩欧拉流的所有局部守恒律与对称性，证明守恒律由经典平衡及广义动量穷尽，对称性均等价于点变换，并给出伴随对称性的乘子刻画。
AI 中文摘要
我们对三维齐次不可压缩欧拉流在速度与压力变量下的所有光滑局部守恒律进行了分类，允许显式依赖时间、位置以及任意有限阶导数。模去在解上为零的流以及散度恒为零的流之后，经典的质量、动量、角动量、动能和螺旋度平衡，连同广义的含时动量，构成了完整的分类。我们还对所有光滑有限阶局部对称特征和伴随对称性进行了分类：它们分别是满足线性化欧拉方程及其在解上的形式伴随的微分表达式。每个对称特征在解上等价于由点变换诱导的对称特征。每个伴随对称性都有一个乘子代表，该乘子与欧拉方程的乘积恒等于某个散度。这些代表穷尽了所有局部守恒律乘子在解上的限制。守恒律的证明分为三个阶段：利用确定方程对伴随对称性进行分类，验证显式乘子恒等式，以及乘子限制与流类之间的对应关系。对最高阶导数响应的共同分析将两个对称性问题都降阶至最多一阶。
英文摘要
We classify all smooth local conservation laws of three-dimensional homogeneous incompressible Euler flow in velocity and pressure, allowing explicit time and position and derivatives of arbitrary finite order. Modulo currents vanishing on solutions and currents with identically zero divergence, the classical mass, momentum, angular-momentum, kinetic-energy and helicity balances, together with generalised time-dependent momentum, exhaust the classification. We also classify all smooth finite-order local symmetry characteristics and adjoint symmetries: differential expressions satisfying the linearised Euler equations and their formal adjoint on solutions, respectively. Every symmetry characteristic is equivalent on solutions to one induced by a point transformation. Every adjoint symmetry has a multiplier representative, whose product with the Euler equations is identically a divergence. These representatives exhaust the restrictions of all local conservation-law multipliers to solutions. The conservation-law proof has three stages: classification of adjoint symmetries using their determining equations, verification of explicit multiplier identities, and the correspondence between multiplier restrictions and current classes. A common analysis of highest-derivative responses reduces both symmetry problems to order at most one.
