二维区域中涡斑的$L^p$稳定性
$L^p$ Stability of Vortex Patches in Two Dimensional Domains
AI总结：
该研究拓展惩罚能量变分框架，在三类典型二维区域上建立了无需先验界的涡斑统一$L^p$稳定性理论，证明极小元满足对应椭圆方程且具有轨道稳定性，显著拓展了可证明稳定的涡结构范围。
AI中文摘要：
本文研究二维不可压缩欧拉方程中涡斑的轨道稳定性，拓展了Abe与Choi为Lamb偶极子开创的惩罚能量变分框架。Abe、Choi与Jeong近期移除$L^1$约束的工作，以及Dong与Luo针对缺乏标度或平移不变性区域的研究，均留下了一个待解决的挑战：在二维区域上建立无需先验$L^1$或$L^p$界的统一$L^p$稳定性理论。我们在三类典型二维区域上建立了统一的$L^p$稳定性理论，这三类区域分别为：半平面、任意宽度的带状区域，以及满足弱有限体积条件的区域。针对每类区域，我们证明了惩罚能量泛函在合适的$p$值下存在极小元，且所有此类极小元均满足椭圆方程$ω^{p-1} = λ(ψ- W x_2)_+$。此外，我们证明极小元集合在欧拉动力学下具有轨道稳定性。由于缺乏空间标度不变性与水平平移不变性，我们需要采用新的策略：在带状区域上，我们改进了集中紧性论证以证明严格次可加性；在弱有限体积区域上，我们通过利用区域固有的衰减速率$q$来保证紧性，从而绕过了次可加性的要求。本工作将多项已有研究的方法整合为一个综合框架，显著拓展了可证明稳定的涡结构的范围。
英文摘要：
In this paper, we investigate the orbital stability of vortex patches in the two-dimensional incompressible Euler equations, extending the penalized energy variational framework pioneered by Abe and Choi \cite{abe2022stability} for Lamb dipoles. The recent work by Abe, Choi and Jeong \cite{Abe2025StabilityOL} (which removes $L^1$ constraint) and Dong and Luo \cite{Dong2026StabilityOV} (which treats domains lacking scaling or translation invariance) left open the challenge of a unified $L^p$ stability theory without any a priori $L^1$ or $L^p$ bounds on two-dimensional domains. We establish a unified $L^p$ stability theory on three typical two-dimensional domains. These domains are: the half-plane, strips of any width, and domains satisfying a weak finite volume condition. For each domain, we prove that the penalized energy functional admits a minimizer for suitable $p$, and that every such minimizer satisfies the elliptic equation $ω^{p-1} = λ(ψ- W x_2)_+$. Furthermore, we demonstrate that the set of minimizers is orbitally stable under the Eulerian dynamics. The absence of spatial scaling and horizontal translation invariance necessitates novel strategies: on the strip, we refine a concentration-compactness argument to prove strict subadditivity; on weak finite volume domains, we bypass the need for subadditivity by exploiting the inherent decay rate $q$ of the domain to enforce compactness. This work synthesizes the approaches of \cite{abe2022stability}, \cite{Abe2025StabilityOL}, \cite{abe2025existence}, and \cite{Dong2026StabilityOV} into a comprehensive framework, significantly expanding the scope of provably stable vortex structures.
