不满足有限质量条件下，奇对称且非负初始扰动对应的Lamb偶极子的稳定性
Stability of Lamb dipoles for odd-symmetric and non-negative initial disturbances without the finite mass condition
AI总结：
本文针对二维Euler方程的Lamb偶极子，在初始涡量扰动满足奇对称、上半平面非负且无有限质量假设的条件下，借助改进能量不等式的新变分表征证明了其稳定性，为二维湍流涡偶极子研究提供理论支撑。
摘要：
受二维湍流中大涡偶极子形成相关实验工作的启发，本文研究了$\n\mathbb{R}^{2}$中二维Euler方程的Lamb偶极子解的稳定性，并探讨Lamb偶极子在何种初始扰动下具备稳定性。我们假设涡量的初始扰动满足(O)关于$x_2$变量的奇对称性，以及(N)在上半平面非负的条件，在不假设(F)有限质量条件的情况下，建立了Lamb偶极子的稳定性定理。证明过程基于利用改进的能量不等式得到的Lamb偶极子的新变分表征。
英文摘要：
In this paper, we consider the stability of the Lamb dipole solution of the two-dimensional Euler equations in $\mathbb{R}^{2}$ and question under which initial disturbance the Lamb dipole is stable, motivated by experimental work on the formation of a large vortex dipole in two-dimensional turbulence. We assume (O) odd symmetry for the $x_2$-variable and (N) non-negativity in the upper half plane for the initial disturbance of vorticity, and establish the stability theorem of the Lamb dipole without assuming (F) finite mass condition. The proof is based on a new variational characterization of the Lamb dipole using an improved energy inequality.
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