AI 中文总结
该研究针对无限通道中二维不可压缩欧拉方程的定态解，分别在光滑与解析范畴分析其灵活性与刚性，发现两类范畴的特性存在差异且解析范畴的稠密刚性具有尖锐性。
AI 中文摘要
我们研究无限通道中二维不可压缩欧拉方程的定态解，其远场极限为均匀非滞止剪切流。在光滑范畴内，对于一大类给定的远场剪切剖面，通过极小极大方法构造流函数的半线性椭圆方程二维解，可得到非剪切定态解。在解析范畴内，我们建立了解析定态的比较原理，并证明对于稠密的解析均匀非滞止剪切剖面族，每个具有给定远场的解析定态本身必为剪切流。特别地，存在远场剪切剖面在光滑范畴表现出灵活性，但在解析范畴表现出刚性；此外，该稠密刚性是尖锐的，因为存在解析剪切剖面在解析范畴内允许灵活性。
英文摘要
We study steady solutions to the two-dimensional incompressible Euler equations in an infinite channel, whose far-field limits are uniformly non-stagnant shear flows. In the smooth category,for a broad class of prescribed far-field shear profiles, non-shear steady states exist via the construction of two-dimensional solutions of the semilinear elliptic equations of stream function by the min--max method. In the analytic category, we establish a comparison principle for the analytic steady states and we show for a dense family of analytic uniformly non-stagnant shear profiles, every analytic steady state with the prescribed far field must itself be a shear flow. In particular, there are far-field shear profiles which exhibit flexibility in the smooth category but rigidity in the analytic category. Furthermore, the dense rigidity is sharp in the sense that there exists analytic shear profile which admits flexibility in the analytic category.
