We study explicit solutions to the 2 dimensional Euler equations in the Lagrangian framework. All known solutions have been of the separation of variables type, where time and space dependence are treated separately. The first such solutions were known already in the 19th century. We show that all the solutions known previously belong to two families of solutions and introduce three new families of solutions. It seems likely that these are all the solutions that are of the separation of variables type.

中文翻译：

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不可压缩的二维Euler方程变量类型分离的显式解

我们研究拉格朗日框架中二维Euler方程的显式解。所有已知的解决方案都是变量类型的分离，其中时间和空间的依赖关系被分别处理。最早的此类解决方案在19世纪就已为人所知。我们证明了以前已知的所有解决方案都属于两个解决方案家族，并介绍了三个新的解决方案家族。这些似乎都是变量分离类型的所有解决方案。