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Hypersonic limit of two-dimensional steady compressible Euler flows passing a straight wedge.(English)Zbl 07800081

Summary: We formulate a problem on hypersonic limit of two-dimensional steady non-isentropic compressible Euler flows passing a straight wedge. It turns out that the Mach number of the upcoming uniform supersonic flow increases to infinity may be taken as that the adiabatic exponent \(\gamma\) of the polytropic gas decreases to 1. We propose a form of the Euler equations which is valid if the unknowns are Radon measures and construct a measure solution containing Dirac measures supported on the surface of the wedge. It is proved that as \(\gamma \to 1\), the sequence of solutions of the compressible Euler equations that contains a shock ahead of the wedge converges vaguely as measures to the measure solution constructed. This justifies the Newton theory of hypersonic flow passing obstacles in the case of two-dimensional straight wedges. The result also demonstrates the necessity of considering general measure solutions in the study of boundary-value problems of systems of hyperbolic conservation laws.

MSC:

35Lxx Hyperbolic equations and hyperbolic systems
35Qxx Partial differential equations of mathematical physics and other areas of application
76Nxx Compressible fluids and gas dynamics

Cite

References:

[1]Anderson, J. D. Jr., Hypersonic and High‐Temperature Gas Dynamics, 2006, AIAA
[2]Brenier, Y.; De Lellis, C.; Székelyhidi, L. Jr., Weak‐strong uniqueness for measure‐valued solutions, Comm. Math. Phys., 305, 351, 2011 ·Zbl 1219.35182
[3]\(B \breve{\operatorname{r}}\) ezina, J.; Feireisl, E., Measure‐valued solutions to the complete Euler system revisited, Z. Angew. Math. Phys., 69, 2018 ·Zbl 1394.35336
[4]Cavalletti, F.; Sedjro, M.; Westdickenberg, M., A variational time discretization for compressible Euler equations, Trans. Amer. Math. Soc., 371, 5083, 2019 ·Zbl 1501.35287
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[8]Hayes, W. D.; Probstein, R. F., Hypersonic Inviscid Flow, 2004, Dover Publications ·Zbl 0148.21502
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[12]Kong, D.‐X.; Wei, C.; Zhang, Q., Formation of singularities in one‐dimensional Chaplygin gas, J. Hyperbolic Differ. Equ., 11, 521, 2014 ·Zbl 1310.35154
[13]Louie, K.; Ockendon, J. R., Mathematical aspects of the theory of inviscid hypersonic flow, Philos. Trans. Roy. Soc. London Ser. A, 335, 121, 1991 ·Zbl 0726.76059
[14]Qu, A.; Yuan, H.; Zhao, Q. ·Zbl 1432.76222
[15]Ruban, A. I., Fluid Dynamics, Part 2: Asymptotic Problems of Fluid Dynamics, 2015, Oxford University Press ·Zbl 1333.76001
[16]Ruban, A. I.; Gajjar, J. S. B., Fluid Dynamics, Part 1: Classical Fluid Dynamics, 2014, Oxford University Press ·Zbl 1298.76001
[17]Sheng, W.; Wang, G.; Yin, G., Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes, Nonlinear Anal. Real World Appl., 22, 115, 2015 ·Zbl 1308.35193
[18]Wang, Z.; Zhang, Y., Steady supersonic flow past a curved cone, J. Differ. Equ., 247, 1817, 2009 ·Zbl 1180.35354
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[20]Zhang, Y.; Zhang, Y., The Riemann problem and interaction of waves in two‐dimesnional steady zero‐pressure adiabatic flow, International Journal of Nonlinear Mechanics, 104, 100, 2018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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