三维非定常势流中定常平面正激波结构的持续性
Persistence of the steady planar normal shock structure in 3-D unsteady potential flows
摘要：
本文研究了非定常势流中垂直于平面固体壁面的平面正激波阵面的定常三维波结构的动力稳定性。稳定性问题可以表述为激波阵面与固壁之间的二面空间区域中的二阶拟线性双曲型方程的自由边界问题。计算的关键困难是由空间域的边界奇异性、激波阵面与固壁的相交曲线引起的。与二维情况下边界的奇异部分只是一个点不同，本文中的三维情况下边界的奇异部分是一条曲线。这种差异给稳定性问题的数学分析带来了新的困难。引入了一种改进的部分速端图变换，使得针对二维情形的延拓技术可以用于建立二面角空间域上二阶线性化双曲型方程初边值问题的适定理论.此外，本文还对延拓技术进行了改进，使得激波阵面的先验估计不会失去规律性。从而构造了经典的非线性迭代格式，证明了稳定性问题解的存在性，从而在不使用Nash-Moser迭代法的情况下，证明了定常平面正激波的动力稳定性。
英文摘要：
This paper concerns the dynamic stability of the steady 3-D wave structure of a planar normal shock front intersecting perpendicularly to a planar solid wall for unsteady potential flows. The stability problem can be formulated as a free boundary problem of a quasi-linear hyperbolic equation of second order in a dihedral-space domain between the shock front and the solid wall. The key difficulty is brought by the edge singularity of the space domain, the intersection curve between the shock front and the solid wall. Different from the 2-D case, for which the singular part of the boundary is only a point, it is a curve for the 3-D case in this paper. This difference brings new difficulties to the mathematical analysis of the stability problem. A modified partial hodograph transformation is introduced such that the extension technique developed for the 2-D case can be employed to establish the well-posed theory for the initial-boundary value problem of the linearized hyperbolic equation of second order in a dihedral-space domain. Moreover, the extension technique is improved in this paper such that loss of regularity in the a priori estimates on the shock front does not occur. Thus the classical nonlinear iteration scheme can be constructed to prove the existence of the solution to the stability problem, which shows the dynamic stability of the steady planar normal shock without applying the Nash-Moser iteration method.
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