A free boundary problem for a quasi‐linear degenerate elliptic equation: Regular reflection of weak shocks
A free boundary problem for a quasi‐linear degenerate elliptic equation: Regular reflection of weak shocks
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DOI：
10.1002/cpa.10013
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发表时间：
2002-01
影响因子：
3
通讯作者：
S. Čanić;B. Keyfitz;Eun Heui Kim
中科院分区：
文献类型：
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作者：
S. Čanić;B. Keyfitz;Eun Heui Kim
本文证明了非定常跨音速小扰动（UTSD）模型中激波被楔形物反射的弱正则反射问题解的存在性。在弱的规则反射中，反射激波后面的状态是超音速的和恒定的.在更下游处，流动变为亚音速;自相似坐标系中的方程在音速线处退化。反射激波变为跨音速，并在那里开始弯曲;它的位置就是退化方程自由边界问题的解。利用Rankine-Hugoniot条件沿着反射激波，我们导出了跨音速激波的演化方程和未知激波位置的斜导数边界条件。通过正则化退化问题，我们构造统一的界限，我们应用局部紧性参数来提取解决问题的极限。该解在内部是光滑的，并且直到退化边界是连续的。
We prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance (UTSD) model for shock reflection by a wedge. In weak regular reflection, the state immediately behind the reflected shock is supersonic and constant. The flow becomes subsonic further downstream; the equation in self‐similar coordinates is degenerate at the sonic line. The reflected shock becomes transonic and begins to curve there; its position is the solution to a free boundary problem for the degenerate equation. Using the Rankine‐Hugoniot conditions along the reflected shock, we derive an evolution equation for the transonic shock, and an oblique derivative boundary condition at the unknown shock position. By regularizing the degenerate problem, we construct uniform bounds; we apply local compactness arguments to extract a limit that solves the problem. The solution is smooth in the interior and continuous up to the degenerate boundary.