Orbital Stability of First Laplacian Eigenstates for the Euler Equation on Flat 2-Tori
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909763117252608 |
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| author | Wang, Guodong |
| author_facet | Wang, Guodong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00750 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orbital Stability of First Laplacian Eigenstates for the Euler Equation on Flat 2-Tori Wang, Guodong Analysis of PDEs 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. |
| title | Orbital Stability of First Laplacian Eigenstates for the Euler Equation on Flat 2-Tori |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.00750 |