Two-dimensional pseudo-steady supersonic flow around a sharp corner for the generalized Chaplygin gas

Aidi Yao; Aidi Yao

doi:10.3934/math.2022654

AIMS Mathematics

2022, Volume 7, Issue 7: 11732-11758. doi: 10.3934/math.2022654

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Research article

Two-dimensional pseudo-steady supersonic flow around a sharp corner for the generalized Chaplygin gas

Aidi Yao ^,

School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, Anhui, 232001, China

Received: 03 July 2021 Revised: 23 November 2021 Accepted: 25 November 2021 Published: 18 April 2022
MSC : 35L65, 35J70, 35R35, 35J65

In this paper, the expansion problem which arises in a two-dimensional (2D) isentropic pseudo-steady supersonic flow expanding into vacuum around a sharp corner for the generalized Chaplygin gas is studied. This expanding problem catches the interaction of an incomplete centered simple wave with a backward planar rarefaction wave and the interaction of a non-planar simple wave with a rigid wall boundary of the 2D self-similar Euler equations. Using the methods of characteristic decompositions and invariant regions, we get the hyperbolicity in the wave interaction domains and prior $ C^{1} $ estimates of solutions to the two interaction problems. It follows the global existence of the solution up to infinity of the gas expansion problem.
Citation: Aidi Yao. Two-dimensional pseudo-steady supersonic flow around a sharp corner for the generalized Chaplygin gas[J]. AIMS Mathematics, 2022, 7(7): 11732-11758. doi: 10.3934/math.2022654

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Abstract

In this paper, the expansion problem which arises in a two-dimensional (2D) isentropic pseudo-steady supersonic flow expanding into vacuum around a sharp corner for the generalized Chaplygin gas is studied. This expanding problem catches the interaction of an incomplete centered simple wave with a backward planar rarefaction wave and the interaction of a non-planar simple wave with a rigid wall boundary of the 2D self-similar Euler equations. Using the methods of characteristic decompositions and invariant regions, we get the hyperbolicity in the wave interaction domains and prior $ C^{1} $ estimates of solutions to the two interaction problems. It follows the global existence of the solution up to infinity of the gas expansion problem.

References

[1]	S. Bang, Interaction of three and four rarefaction waves of the pressure-gradient system, J. Differ. Equations, 246 (2009), 453–481. https://doi.org/10.1016/j.jde.2008.10.001 doi: 10.1016/j.jde.2008.10.001
[2]	T. Chang, G. Q. Chen, S. L. Yang, On the 2-D Riemann problem for the compressible Euler equations Ⅰ. Interaction of shocks and rarefaction waves, Discrete Cont. Dyn.-A, 1 (1995), 555–584.
[3]	S. X. Chen, A. F. Qu, The two-dimensional Riemann prolem for Chaplygin gas, SIAM J. Math. Anal., 44 (2012), 2146–2178. https://doi.org/10.1137/110838091 doi: 10.1137/110838091
[4]	X. Chen, Y. X. Zheng, The interaction of rarefaction waves of the two-dimensional Euler equations, Indiana Univ. Math. J., 59 (2010), 231–256. https://doi.org/10.1512/iumj.2010.59.3752 doi: 10.1512/iumj.2010.59.3752
[5]	R. Courant, K. O. Friedrichs, Supersonic flow and shock waves, Berlin-Heidelberg-New York. Springer-Verlag, 1976.
[6]	Z. H. Dai, T. Zhang, Existence of a global smooth solution of a degenertate Goursat problem of gas dynamics, Arch. Ration. Mech. Anal., 155 (2000), 277–298.
[7]	J. Ge, W. C. Sheng, The two dimensional gas expansion prolem of the Euler equations for the generalized Chaplygin gas, Comm. Pure Appl. Anal., 13 (2014), 2733–2748. https://doi.org/10.3934/cpaa.2014.13.2733 doi: 10.3934/cpaa.2014.13.2733
[8]	L. H. Guo, W. C. Sheng, T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Comm. Pure Appl. Anal., 9 (2010), 431–458.
[9]	G. Lai, On the expansion of a wedge of van der Waals gas into a vacuum, J. Differ. Equations, 259 (2015), 1181–1202. https://doi.org/10.1016/j.jde.2015.02.039 doi: 10.1016/j.jde.2015.02.039
[10]	G. Lai, On the expansion of a wedge of van der Waals gas into a vacuum Ⅱ, J. Differ. Equations, 260 (2016), 3538–3575. https://doi.org/10.1016/j.jde.2015.10.048 doi: 10.1016/j.jde.2015.10.048
[11]	G. Lai, Global solutions to a class of two-dimensional Riemann problem for the insentropic Euler equations with a general equation of state, Indiana Univ. Math. J., 68 (2019), 1409–1464. https://doi.org/10.1512/iumj.2019.68.7782 doi: 10.1512/iumj.2019.68.7782
[12]	G. Lai, C. Shen, Characteristic decompositions and boundary value problems for two-dimensional steady relativistic Euler equations, Math. Meth. Appl. Sci., 37 (2014), 136–147. https://doi.org/10.1002/mma.2791 doi: 10.1002/mma.2791
[13]	G. Lai, W. C. Sheng, Centered wave bubbles with sonic boundary of pseudosteady Guderley Mach refection configuration in gas dynamics, J. Math. Pures Appl., 104 (2015), 179–206. https://doi.org/10.1016/j.matpur.2015.02.005 doi: 10.1016/j.matpur.2015.02.005
[14]	G. Lai, W. C. Sheng, Elementary wave interactions to the compressible Euler equations for Chaplygin gas in two dimensions, SIAM J. Appl. Math., 76 (2016), 2218–2242. https://doi.org/10.1137/16M1061801 doi: 10.1137/16M1061801
[15]	G. Lai, W. C. Sheng, Two-dimensional pseudosteady flows around a sharp corner, Arch. Ration. Mech. An., 241 (2021), 805–884. https://doi.org/10.1007/s00205-021-01665-0 doi: 10.1007/s00205-021-01665-0
[16]	G. Lai, W. C. Sheng, Y. X. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions, Disc. Cont. Dyna. Syst., 31 (2011), 489–523.
[17]	P. D. Lax, X. D. Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comp., 19 (2006), 319–340.
[18]	L. E. Levine, The expansion of a wedge of gas into a vacuum, Proc. Camb. Philol. Soc., 64 (1968), 1151-1163. https://doi.org/10.1017/S0305004100043899 doi: 10.1017/S0305004100043899
[19]	J. Q. Li, On the two-dimensional gas expansion for compressible Euler eqautions, SIAM J. Appl. Math., 62 (2001), 831–852. https://doi.org/10.1137/S0036139900361349 doi: 10.1137/S0036139900361349
[20]	J. Q. Li, W. C. Sheng, T. Zhang, Two dimensional Riemann problems: From scalar conservation laws to compressible Euler equations, Acta. Math. Sci., 29 (2009), 777–802. https://doi.org/10.1016/S0252-9602(09)60070-9 doi: 10.1016/S0252-9602(09)60070-9
[21]	J. Q. Li, Z. C. Yang, Y. X. Zheng, Characteristic decompositions and interactions of rarefaction waves of 2-d Euler equations, J. Differ. Equations, 250 (2011), 782–798. https://doi.org/10.1016/j.jde.2010.07.009 doi: 10.1016/j.jde.2010.07.009
[22]	J. Q. Li, T. Zhang, Y. X. Zheng, Simple waves and a characteristics decomposition of the two-dimensional compressible Euler equations, Comm. Math. Phys., 267 (2006), 1–12.
[23]	J. Q. Li, Y. X. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Rational Mech. Anal., 193 (2009), 623–657. https://doi.org/10.1007/s00205-008-0140-6 doi: 10.1007/s00205-008-0140-6
[24]	J. Q. Li, Y. X. Zheng, Interaction of four Rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations, Comm. Math. Phys., 296 (2010), 303–321.
[25]	T. T. Li, W. C. Sheng, The general Riemann problem for the linearized system of two-dimensional isentropic flow in gas dynamics, J. Math. Anal. Appl., 276 (2002), 598–610. https://doi.org/10.1016/S0022-247X(02)00315-3 doi: 10.1016/S0022-247X(02)00315-3
[26]	T. T. Li, W. C. Yu, Boundary value problem for quasilinear hyperbolic systems, Duke University, 1985.
[27]	W. C. Sheng, Two-dimensional Riemann problem for scalar conservation laws, J. Differ. Equations, 183 (2002), 239–261. https://doi.org/10.1006/jdeq.2001.4124 doi: 10.1006/jdeq.2001.4124
[28]	W. C. Sheng, G. D. Wang, T. Zhang, Critical transonic shock and supersonic bubble in oblique rarefaction wave reflection along a compressive corner, SIAM J. Appl. Math., 70 (2010), 3140–3155. https://doi.org/10.1137/090760362 doi: 10.1137/090760362
[29]	A. D. Yao, W. C. Sheng, Two-dimensional pseudo-steady supersonic flow around a sharp corner, Z. Angew. Math. Mech., 102 (2022). https://doi.org/10.1002/zamm.201800270 doi: 10.1002/zamm.201800270
[30]	W. C. Sheng, A. D. Yao, Centered simple waves for the two-dimensional pseudo-steady isothermal flow around a convex corner, Appl. Math. Mech., 40 (2019), 705–718. https://doi.org/10.1007/s10483-019-2475-6 doi: 10.1007/s10483-019-2475-6
[31]	W. C. Sheng, S. K. You, Interaction of a centered simple wave and a planar rarefacion wave of the two-dimensional Euler equations for pseudo-steady compresssible flow, J. Math. Pure. Appl., 114 (2018), 29–50. https://doi.org/10.1016/j.matpur.2017.07.019 doi: 10.1016/j.matpur.2017.07.019
[32]	V. A. Suchkow, Flow into a vacuum along an oblique wall, J. Appl. Math. Mech., 27 (1963), 1132–1134. https://doi.org/10.1016/0021-8928(63)90195-3 doi: 10.1016/0021-8928(63)90195-3
[33]	M. N. Sun, C. Shen, On the Riemann problem for 2-D compressible Euler equations in three piece, Nonlinear Anal., 70 (2009), 3773–3780. https://doi.org/10.1016/j.na.2008.07.033 doi: 10.1016/j.na.2008.07.033
[34]	G. D. Wang, B. C. Chen, Y. B. Hu, The two-dimensional Riemann problem for Chaplygin gas dynamics with three constant states, J. Math. Anal. Appl., 393 (2012), 544–562.
[35]	T. Zhang, Y. X. Zheng, Conjecture on the structure of solution of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Math. Anal., 21 (1990), 593–630. https://doi.org/10.1137/0521032 doi: 10.1137/0521032
[36]	W. X. Zhao, The expansion of gas from a wedge with small angle into a vacuum, Comm. Pure Appl. Anal., 12 (2013), 2319–1330.

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