Hypersonic Similarity and Thin Shock Layer Theory for Two-Dimensional Steady Euler Flows

Project: Research

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The aim of this project is to develop the rigorous mathematical theory on the validation of the hypersonic similarity, that is also called the Mach-number independence principle, as well as the mathematical analysis on the thin shock layer problem. Particularly, it willmainly focus on the problem of the two dimensional steady compressible hypersonic full Euler flows over a curved slender wedge, and other related problems.The hypersonic flow is the flow whose Mach number is bigger than five. As said in [2,pp.12], it is physically different from the  upersonic flow, and important due to the hypersonic vehicle designs for the 21st century. The study of the hypersonic flow is challenging, because of the phenomena of the thin shock layer, entropy layer, high-temperature flow, and low density flow, etc.. Therefore, for the mathematical theory of the hypersonic flow, we need to study the weak solutions with free boundary in a very narrow cornered domain, where the density could be small (like the flows near the vacuum), the entropy gradients could be large, or the equations for the reaction and radiation process could be taken into account. In this project, we will select one of the typical problems discussed in [2], that is the mathematical theory of the compressible Euler flow over a curved slender wedge. More precisely, we willwork on the rigorous proof of the validation of the hypersonic similarity for the compressible full Euler flow with shock waves, athematical analysis on the thin shock layer problem, and then study the problem for other related models such as the exothermically reacting Euler equations and the radiative Euler equations.Since Courant-Friedriches’ classic book [16], relatively satisfactory mathematical results are established for the supersonic steady Euler equations past a two-dimensional wedge or a cone (cf. [6, 7, 9–14, 20, 24–26, 30, 32–34]). However, the rigorous analysis for the hypersonic flow is far from satisfactory, even though a lot of literatures on the hypersonic flow are available since 1940s (cf. [28, 29]). Due to the important applications in the hypersonic areodynamics, it is valuable for us to study the hypersonic flows by working on the objectives proposed in this project based on the techniques developped for the steady supersonic flow problems. We have been working on the related problems for several years. We proved the hypersonic similarity for the two dimensional compressible irrotational Euler flows in [22]; validation of the weakly nonlinear geometric optics for one-dimensional hyperbolic systems of conservation laws in [8]; asymptotic stability of the shock waves or the rarefaction waves for the radiative Euler equations in [18, 19]; and the global existence of steady supersonic exothermically reacting Euler flows with strong contact discontinuity over a Lipschitz wall in [31]. In this project, we will mainly focus on a further development of the technics of proving the obtained results, to study the mentioned objectives.


Project number9043031
Grant typeGRF
Effective start/end date1/01/21 → …