无限长喷嘴中具有驻点的定常 Euler 流分析
Analysis on the steady Euler flows with stagnation points in an infinitely long nozzle
AI总结：
本文研究无限长喷嘴中含驻点的定常不可压缩 Euler 流，证明上游为 Poiseuille 流时的全局唯一性与解的存在性，并建立非驻点区域障碍型自由边界的全局 $C^1$ 正则性及驻点区域存在条件。
摘要：
近期一项重要结果断言，在二维无限长条带中，严格远离驻点的定常不可压缩 Euler 流必定是剪切流。另一方面，具有驻点的流在分析上极具挑战性，同时也是流体力学中有趣且重要的现象。本文不仅证明了具有驻点的定常流的唯一性与存在性，还得到了驻点集边界的正则性；该边界是一类障碍型自由边界。首先，在无限长条带中，针对以上游远场状态为 Poiseuille 流的定常 Euler 方程组，我们证明了一个全局唯一性定理。由于驻点的出现，流函数所满足的半线性方程的非线性变为非 Lipschitz 的。这带来了一个具有挑战性的分析问题，因为许多经典分析方法无法直接应用。其次，通过变分方法，我们在无限长喷嘴中建立了在上游趋于 Poiseuille 流的定常不可压缩 Euler 流的存在性。一个非常有趣的现象是非驻点区域边界的正则性：该边界可视为障碍型自由边界，并被证明具有全局 $C^1$ 正则性。最后，只要喷嘴宽度大于上游处（即流趋于 Poiseuille 流之处）喷嘴的宽度，就证明了驻点区域的存在性。
英文摘要：
A recent prominent result asserts that steady incompressible Euler flows strictly away from stagnation in a two-dimensional infinitely long strip must be shear flows. On the other hand, flows with stagnation points, very challenging in analysis, are interesting and important phenomenon in fluids. In this paper, we not only prove the uniqueness and existence of steady flows with stagnation points, but also obtain the regularity of the boundary of stagnation set, which is a class of obstacle type free boundary. First, we prove a global uniqueness theorem for steady Euler system with Poiseuille flows as upstream far field state in an infinitely long strip. Due to the appearance of stagnation points, the nonlinearity of the semilinear equation for the stream function becomes non-Lipschitz. This creates a challenging analysis problem since many classical analysis methods do not apply directly. Second, the existence of steady incompressible Euler flows, tending to Poiseuille flows in the upstream, are established in an infinitely long nozzle via variational approach. A very interesting phenomenon is the regularity of the boundary of non-stagnant region, which can be regarded as an obstacle type free boundary and is proved to be globally $C^1$. Finally, the existence of stagnation region is proved as long as the nozzle is wider than the width of the nozzle at upstream where the flows tend to Poiseuille flows.
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