Stability of transonic shock for supersonic flow past a wedge. (English) Zbl 1348.35139
The authors consider steady 2D transonic polytropic flow hitting a wedge. The Euler system describes the flow with enthalpy. The wedge is curved slightly compared to a straight wedge. Using weighted Hölder norms, the authors show that the shock solution is structurally stable.
Reviewer: Ilya A. Chernov (Petrozavodsk)
MSC:
| 35L67 | Shocks and singularities for hyperbolic equations |
| 35L65 | Hyperbolic conservation laws |
| 35R35 | Free boundary problems for PDEs |
| 35B35 | Stability in context of PDEs |
| 35Q31 | Euler equations |
References:
| [1] | CourantR, FriedrichsKO. Supersonic Flow and Shock Waves. Interscience Publishers: New York, 1948. · Zbl 0041.11302 |
| [2] | ChenSX, FangBX. Stability of transonic shocks in supersonic flow past a wedge. Journal of Differential Equations.2007; 233:105-135. DOI: 10.1016/j.jde.2006.09.020. · Zbl 1114.35126 |
| [3] | FangBX. Stability of transonic shocks for the full Euler system in supersonic flow past a wedge. Mathematical Methods in the Applied Sciences. 2006; 29:1-26. DOI: 10.1002/mma.661. · Zbl 1090.35120 |
| [4] | CanicS, KeyfitzB, LiebermanGM. A proof of existence of perturbed steady transonic shocks via a free boundary problem. Communications on Pure and Applied Mathematics. 2000; 53:484-511. DOI: 10.1002/(SICI)1097‐0312(200004)53:4<484::AID‐CPA3>3.0.CO;2‐K. · Zbl 1017.76040 |
| [5] | CanicS, KeyfitzB, KimEH. A free boundary problem for a quasilinear degenerate elliptic equation: regular reflection of weak shocks. Communications on Pure and Applied Mathematics. 2002; 55:71-92. DOI 10.1002/cpa.10013. · Zbl 1124.35338 |
| [6] | ChenGQ, FeldmanM. Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. Journal of the American Mathematical Society. 2003; 16:464-491. DOI: 10.1090/S0894‐0347‐03‐00422‐3. |
| [7] | ChenSX. Transonic in 3‐D compressible flow passing a duct with a general section for Euler systems. Transactions of the American Mathematical Society. 2008; 360:5265-5289. DOI: 10.1090/S0002‐9947‐08‐04493‐0. · Zbl 1158.35064 |
| [8] | XinZP, YinHC. Transonic shock in a nozzle I, Two‐dimensional case. Communications on Pure and Applied Mathematics. 2005; 58:999-1050. DOI: 10.1002/cpa.20025. · Zbl 1076.76043 |
| [9] | YinHC, ZhouCH. On global transonic shocks for the steady supersonic Euler flows past sharp 2‐D wedges. Journal of Differential Equations. 2009; 246:4466-4496. DOI: 10.1016/j.jde.2008.12.009. · Zbl 1178.35257 |
| [10] | GrisvardP. Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program): Boston, MA, 1985. · Zbl 0695.35060 |
| [11] | DongHJ, KimS. Partial Schauder estimates for second‐order elliptic and parabolic equations. Calculus of Variations and Partial Differential Equations. 2011; 40:481-500. DOI: 10.1007/s00526‐010‐0348‐9. · Zbl 1221.35078 |
| [12] | FifeP. Schauder estimates under incomplete Hölder continuity assumptions. Pacific Journal of Mathematics. 1963; 13:511-550. · Zbl 0121.32101 |
| [13] | TianGJ, WangXJ. Partial regularity for elliptic equations. Discrete and Continuous Dynamical Systems. 2010; 28:899-913. DOI: 10.3934/dcds.2010.28.899. · Zbl 1193.35025 |
| [14] | GilbargD, HörmanderL. Intermediate Schauder estimates. Archive for Rational Mechanics and Analysis. 1980; 74:297-318. DOI: 10.1007/BF00249677. · Zbl 0454.35022 |
| [15] | LiebermanGM. The Perron process applied to oblique derivative problems. Advances in Mathematics. 1985; 55(2): 161-172. DOI: 10.1016/0001‐8708(85)90019‐2. · Zbl 0567.35027 |
| [16] | LiebermanGM. Intermediate Schauder estimates for oblique derivative problems. Archive for Rational Mechanics and Analysis. 1986; 93:129-134. DOI: 10.1007/BF00279956. · Zbl 0603.35025 |
| [17] | LiebermanGM. Oblique derivative problems in Lipschitz domains. I. Continuous boundary data. Bollettino dell’Unione Matematica Italiana. 1987; B(7): 1185-1210. · Zbl 0637.35028 |
| [18] | LiebermanGM. Oblique derivative problems in Lipschitz domains. II. Discontinuous boundary data. Journal fr̈ die reine und angewandte Mathematik. 1988; 389:1-21. DOI: 10.1515/crll.1988.389.1. · Zbl 0648.35033 |
| [19] | GilbargD, TrudingerNS. Elliptic Partial Differential Equations of Second Order. Springer‐Verlag: Berlin, 2001. · Zbl 1042.35002 |
| [20] | ChenSX, HuD, FangBX. Stability of the E‐H type regular shock refraction. Journal of Differential Equations.2013; 254:3146-3199. 10.1016/j.jde.2012.12.018. · Zbl 1272.35152 |
| [21] | ChenSX. Stability of a Mach configuration. Journal of the American Mathematical Society. 2006; 19:1-35. · Zbl 1083.35064 |
| [22] | KozlovVA, Maz’yaVG, RossmannJ. Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, vol. 52. American Mathematical Society: Providence, RI, 1997. · Zbl 0947.35004 |
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