We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different from, and larger than, the family of smooth stationary solutions recently obtained by Gavrilov and Constantin-La-Vicol; in particular, these solutions are not localizable. A key step in the proof is the construction of solutions to an overdetermined elliptic boundary value problem where one prescribes both Dirichlet and (nonconstant) Neumann data.

中文翻译：

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通过超定边界问题具有紧支撑的分段平滑平稳欧拉流

我们构建了具有紧凑支持的 3D Euler 方程的新的平稳弱解。解决方案是分段平滑且在表面上不连续的，它们是带涡旋的轴对称。我们找到的解的范围不同于最近由 Gavrilov 和 Constantin-La-Vicol 获得的平滑平稳解系列，并且更大；特别是，这些解决方案不可本地化。证明中的一个关键步骤是构建超定椭圆边界值问题的解决方案，其中规定了狄利克雷和（非常量）诺依曼数据。