Title: Orbital Stability of First Laplacian Eigenstates for the Euler Equation on Flat 2-Tori Show Chinese title

Title: 第一拉普拉斯特征态在平面上二维环面上欧拉方程的轨道稳定性

Abstract: On a two-dimensional flat torus, the Laplacian eigenfunctions can be expressed explicitly in terms of sinusoidal functions. For a rectangular or square torus, it is known that every first eigenstate is orbitally stable up to translation under the Euler dynamics. In this paper, we extend this result to flat tori of arbitrary shape. As a consequence, we obtain for the first time a family of orbitally stable sinusoidal Euler flows on a hexagonal torus. The proof is carried out within the framework of Burton's stability criterion and consists of two key ingredients: (i) establishing a suitable variational characterization for each equimeasurable class in the first eigenspace, and (ii) analyzing the number of translational orbits within each equimeasurable class. The second ingredient, particularly for the case of a hexagonal torus, is very challenging, as it requires analyzing a sophisticated system of polynomial equations related to the symmetry of the torus and the structure of the first eigenspace. Show Chinese abstract

Abstract: 在二维平面上的环面，拉普拉斯特征函数可以显式地用正弦函数表示。 对于矩形或正方形环面，已知每个第一特征态在欧拉动力学下，相对于平移是轨道稳定的。 在本文中，我们将这一结果扩展到任意形状的平坦环面。 作为结果，我们首次在六边形环面上获得了一族轨道稳定的正弦欧拉流。 证明是在伯顿稳定性准则的框架内进行的，包括两个关键要素：(i) 为第一特征空间中的每个等测类建立合适的变分特征，以及 (ii) 分析每个等测类中的平移轨道数量。 第二个要素，特别是对于六边形环面的情况，非常具有挑战性，因为它需要分析与环面对称性和第一特征空间结构相关的复杂多项式方程系统。