AI 中文总结
该研究针对二维环面上带独立傅里叶系数的高斯初值欧拉方程，证明了对应高斯测度在欧拉流下不变的充要条件是支撑为剪切流或胞腔流，解决了相关分类猜想，证明基于傅里叶支撑封闭性与逆方差仿射关系。
AI 中文摘要
我们研究了二维环面$\u21a4^2$上具有独立傅里叶系数的高斯随机初值的不可压缩欧拉方程。对任意$σ>0$，我们证明了$H^σ(\u21a4^2)$上的这类高斯测度在欧拉流下不变，当且仅当它的支撑集为剪切流或胞腔流。这解决了Bedrossian与Latocca提出的不变测度分类猜想（发表于《Ann. Inst. H. Poincaré C Anal. Non Linéaire》，2026年）。证明依赖于非退化相互作用下傅里叶支撑的封闭性，以及欧拉三元组对应的逆方差之间的仿射关系。
英文摘要
We consider the two-dimensional incompressible Euler equation on $\mathbb T^2$ with Gaussian random initial data having independent Fourier coefficients. For every $σ>0$, we prove that such a Gaussian measure on $H^σ(\mathbb T^2)$ is invariant under the Euler flow if and only if it is supported either on shear flows or on cellular flows. This settles the invariant-measure classification conjecture posed by Bedrossian and Latocca (Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2026). The proof relies on closure of the Fourier support under non-degenerate interactions and an affine relation among the inverse variances along Euler triples.
