直接马赫构型的局部稳定性
Local stability of direct Mach configurations
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中文总结 AI 辅助
本文研究二维定常可压缩欧拉方程中直接马赫构型的局部稳定性，通过拉格朗日坐标和椭圆方程混合边值问题，证明了在横截性条件下小扰动的唯一性。
中文摘要 AI 辅助
我们研究了二维定常可压缩欧拉方程中直接马赫构型的局部稳定性。给定一个由入射激波、反射激波、马赫杆和滑移线组成的分段常数马赫构型，且下游状态为亚音速，我们证明了以下结论：在激波极线环的横截性条件和截断段的适当条件下，来流的任意足够小的扰动都会产生一个马赫构型，该构型是背景构型的小扰动，即反射激波、马赫杆和滑移线以及下游亚音速流都保持接近其背景对应物；此外，在满足先验估计的解类中（对于在稍强范数下小的来流），该解是唯一的。采用拉格朗日坐标将未知的接触间断曲线变换为固定边界，并将完整的欧拉系统简化为关于流动方向和压力的一阶椭圆系统。处理接触间断的关键思想是求解一个具有间断系数的散度形式的混合边值问题，针对单个椭圆方程，使得接触间断条件自然地作为椭圆问题解的相容性条件被保留。
英文摘要
We investigate the local stability of direct Mach configurations for the two-dimensional steady compressible Euler equations. Given a piecewise constant Mach configuration that consists of an incident shock, a reflected shock, a Mach stem and a slip line, with subsonic downstream states, we prove the following: under a transversality condition on the shock polar loops and a suitable condition on the truncation segment, every sufficiently small perturbation of the incoming flow gives rise to a Mach configuration that is a small perturbation of the background one, in the sense that the reflected shock, the Mach stem and the slip line, together with the downstream subsonic flow, stay close to their background counterparts; moreover, the solution is unique in the class of solutions satisfying the a priori estimate (for incoming flows that are small in a slightly stronger norm). Lagrangian coordinates are employed to transform the unknown contact discontinuity curve into a fixed boundary and to reduce the full Euler system to a first-order elliptic system for the flow direction and the pressure. The key idea in dealing with the contact discontinuity is to solve a mixed boundary value problem, in divergence form with discontinuous coefficients, for a single elliptic equation, so that the contact discontinuity conditions are naturally preserved as compatibility conditions for the solutions of the elliptic problem.
发表机构
- China Three Gorges University（三峡大学）
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