AI 中文总结
研究构建三维不可压缩欧拉方程的平稳解，其具有螺旋对称性且支撑在螺旋线邻域。通过与流函数公式相关的非线性超定椭圆边值问题得到分段光滑解，解的涡旋横截面各向异性，关键步骤是分析特定超定椭圆问题。
AI 中文摘要
我们构建了具有螺旋对称性且支撑在螺旋线邻域内的三维不可压缩欧拉方程的平稳解。这些解是分段光滑的，源自与流函数公式相关的非线性超定椭圆边值问题。其显著特征是涡旋横截面本质上是各向异性的：重新缩放后，主导形状是椭圆形而非径向的，边界呈现出非平凡的第三傅里叶模式，反映了先前轴对称构造中不存在的螺旋效应。证明中的关键步骤是分析具有规定狄利克雷和非恒定诺伊曼条件的真正各向异性超定椭圆问题。
英文摘要
We construct stationary solutions of the three-dimensional incompressible Euler equations with helical symmetry and support in a neighborhood of a helix. The solutions are piecewise smooth and arise from a nonlinear overdetermined elliptic boundary value problem associated with a stream-function formulation. A distinguishing feature is that the vortex cross-sections are intrinsically anisotropic: after rescaling, the leading-order shape is elliptic rather than radial, and the boundary exhibits a nontrivial third Fourier mode reflecting helical effects absent in previous axisymmetric constructions. A key step in the proof is the analysis of a genuinely anisotropic overdetermined elliptic problem with prescribed Dirichlet and nonconstant Neumann conditions.
Comments41 pages. arXiv admin note: text overlap with arXiv:2005.04380
