带边界无界区域中Beltrami流的刘维尔型定理

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关键词：Beltrami流；刘维尔型定理；星形域；弱积分条件；无界区域；单位外法向量；截断函数图分类号：0175.2 文献标志码：A doi：10.12415/j.issn.1671-7872.23155

Abstract： Liouvile-type theorems for Beltrami flows in unbounded domains with starlike boundaries were established under weakened integral conditions，yielding four distinct criteria.Two key lemmas were introduced to construct exact identities between volume and surface integrals of the Beltrami flows. The proofs were developed by combining contradiction methods，refined inequality estimates，and geometric properties of the unit outward normal vector on starlike boundaries.Notably，a truncation function technique was employed to addressspecial critical cases.These results were shown to significantly relax the integral requirements in existing literature and provide new theoretical tools for analyzing Beltrami flows in unbounded domains.

Keywords：Beltrami flow;Liouville type theorem; star-shaped domain; weakened integral conditions; unbounded domains; unit outer normal vector; cut-off function

Beltrami流指的是向量值函数 u 满足下面的方程组

可知，每个Beltrami流都是稳态欧拉方程的特解， × 为向量的向量积，·为向量的数量积， 为梯度算子，curl u=∇×u 为函数 u 的旋度，div u=∇⋅u 为函数 u 的散度。（剩余7520字）