发表机构
University of Macau; Hunan University; Shanghai Jiao Tong University（澳门大学; 湖南大学; 上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究二维稳态不可压缩欧拉流，证明有限总曲率流在无驻点时必为平行剪切流，压力在无穷远一致收敛，并给出总曲率下界，同时将刚性结果推广至更广条件并优化正则性假设。
AI 中文摘要
我们研究具有有限总曲率的二维稳态不可压缩欧拉流。此类流的一个显著族源于二维半线性椭圆方程的有限Morse指标解。我们建立了$L^\infty\cap H^1_{\rm loc}$类中有限总曲率流的三个主要性质。首先，在平面和半平面中，每个无驻点的此类流都是平行剪切流。其次，在平面、半平面和无限条带中，压力在无穷远每一端一致收敛于常数。第三，在平面、半平面、无限条带和周期条带中，我们建立了以压力振荡表示的总曲率的锐利下界。刚性结果实际上扩展到更广泛的流类：在平面和半平面中，任何无驻点的稳态欧拉流都是平行剪切流，只要$\nabla P\in L^q$对某个$1\le q\le 2$成立。结合压力的标准椭圆估计，该结果给出了Hamel和Nadirashvili刚性定理的一个简洁新证明，并将正则性假设最优地锐化至$H^1_{\rm loc}$。在整个分析中，压力作为统一量贯穿始终。
英文摘要
We study two-dimensional steady incompressible Euler flows with finite total curvature. A distinguished family of such flows arises from finite-Morse-index solutions to semilinear elliptic equations in two dimensions. We establish three main properties of finite-total-curvature flows in the $L^\infty\cap H^1_{\rm loc}$ class. First, in the plane and the half-plane, every such flow without stagnation points is a parallel shear flow. Second, in the plane, the half-plane, and the infinite strip, the pressure converges uniformly to a constant at each end at infinity. Third, in the plane, the half-plane, the infinite strip, and the periodic strip, we establish sharp lower bounds for the total curvature in terms of the pressure oscillation. The rigidity result in fact extends to a broader class of flows: in the plane and the half-plane, any steady Euler flow without stagnation points is a parallel shear flow provided that $\nabla P\in L^q$ for some $1\le q\le 2$. Combined with standard elliptic estimates for the pressure, this result yields a concise new proof of Hamel and Nadirashvili's rigidity theorems, with the regularity assumption sharpened optimally to $H^1_{\rm loc}$. The pressure serves as a unifying quantity throughout the analysis.
