December  2019, 12(6): 1197-1228. doi: 10.3934/krm.2019046

Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations

1. 

Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China

2. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States

* Corresponding author: Yanbo Hu

Received  June 2018 Revised  May 2019 Published  September 2019

Fund Project: The first author is supported by NSF of Zhejiang Province of China grant LY17A010019, NSFC grants 11301128, 11571088 and China Scholarship Council grant 201708330155.

This paper is focused on the existence of classical sonic-supersonic solutions near sonic curves for the two-dimensional pseudo-steady full Euler equations in gas dynamics. By introducing a novel set of change variables and using the idea of characteristic decomposition, the Euler system is transformed into a new system which displays a transparent singularity-regularity structure. With a choice of weighted metric space, we establish the local existence of smooth solutions for the new system by the fixed-point method. Finally, we obtain a local classical solution for the pseudo-steady full Euler equations by converting the solution from the partial hodograph variables to the original variables.

Citation: Yanbo Hu, Tong Li. Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations. Kinetic and Related Models, 2019, 12 (6) : 1197-1228. doi: 10.3934/krm.2019046
References:
[1]

L. Bers, On the continuation of a potential gas flow across the sonic line, Tech. Notes Nat. Adv. Comm. Aeronaut., 1950 (1950), 58 pp.

[2]

S. ČanićB. L. Keyfitz and E. H. Kim, A free boundary problem for a quasi-linear degenerate elliptic equation: Regular reflection of weak shocks, Comm. Pure Appl. Math., 55 (2002), 71-92.  doi: 10.1002/cpa.10013.

[3]

T. ChangG. Q. Chen and S. L. Yang, On the 2-D Riemann problem for the compressible Euler equations. Ⅰ. Interaction of shocks and rarefaction waves, Discrete Contin. Dyn. Syst., 1 (1995), 555-584.  doi: 10.3934/dcds.1995.1.555.

[4]

G.-Q. ChenC. M. DafermosM. Slemrod and D. H. Wang, On two-dimensional sonic-subsonic flow, Comm. Math. Phys., 271 (2007), 635-647.  doi: 10.1007/s00220-007-0211-9.

[5]

G.-Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc., 16 (2003), 461-494.  doi: 10.1090/S0894-0347-03-00422-3.

[6]

G.-Q. ChenF.-M. Huang and T.-Y. Wang, Subsonic-sonic limit of approximate solutions to multidimensional steady Euler equations, Arch. Ration. Mech. Anal., 219 (2016), 719-740.  doi: 10.1007/s00205-015-0905-7.

[7]

G.-Q. ChenM. Slemrod and D. H. Wang, Vanishing viscosity method for transonic flow, Arch. Ration. Mech. Anal., 189 (2008), 159-188.  doi: 10.1007/s00205-007-0101-5.

[8]

S. X. Chen, Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems, Trans. Amer. Math. Soc., 360 (2008), 5265-5289.  doi: 10.1090/S0002-9947-08-04493-0.

[9]

S. X. Chen, Compressible flow and transonic shock in a diverging nozzle, Comm. Math. Phys., 289 (2009), 75-106.  doi: 10.1007/s00220-009-0811-7.

[10]

S. X. Chen, The fundamental solution of the Keldysh type operator, Sci. China Ser. A, 52 (2009), 1829-1843.  doi: 10.1007/s11425-009-0069-8.

[11]

S. X. ChenZ. P. Xin and H. C. Yin, Global shock waves for the supersonic flow past a perturbed cone, Comm. Math. Phys., 228 (2002), 47-84.  doi: 10.1007/s002200200652.

[12]

X. Chen and Y. X. Zheng, The direct approach to the interaction of rarefaction waves of the two-dimensional Euler equations, Indiana Univ. Math. J., 59 (2010), 231-256.  doi: 10.1512/iumj.2010.59.3752.

[13]

L. P. Cook, Transonic Aerodynamics: Problems in Asymptotic Theory, Frontiers in Applied Mathematics, 12. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1993. doi: 10.1137/1.9781611970975.

[14]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y. 1948.

[15]

Z. H. Dai and T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Ration. Mech. Anal., 155 (2000), 277-298.  doi: 10.1007/s002050000113.

[16]

V. Elling and T.-P. Liu, Supersonic flow onto a solid wedge, Comm. Pure Appl. Math., 61 (2008), 1347-1448.  doi: 10.1002/cpa.20231.

[17]

C. Ferrari and F. Tricomi, Transonic Aerodynamics, Academic Press, 1968.

[18]

K. G. Guderley, The Theory of Transonic Flow, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962.

[19]

Y. B. Hu and J. Q. Li, Sonic-supersonic solutions for the two-dimensional steady full Euler equations, submitted, 2017.

[20]

Y. B. HuJ. Q. Li and W. C. Sheng, Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations, Z. Angew. Math. Phys., 63 (2012), 1021-1046.  doi: 10.1007/s00033-012-0203-2.

[21]

Y. B. Hu and T. Li, The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations, Comm. Pure Appl. Anal., 18 (2019), 3317-3336.  doi: 10.3934/cpaa.2019149.

[22]

Y. B. Hu and G. D. Wang, Semi-hyperbolic patches of solutions to the two-dimensional nonlinear wave system for Chaplygin gases, J. Differential Equations, 257 (2014), 1567-1590.  doi: 10.1016/j.jde.2014.05.020.

[23]

A. G. Kuz'min, Boundary Value Problems for Transonic Flow, John Wiley and Sons, 2002.

[24]

G. Lai and W. C. Sheng, Centered wave bubbles with sonic boundary of pseudosteady Guderley Mach reflection configurations in gas dynamics, J. Math. Pure Appl., 104 (2015), 179-206.  doi: 10.1016/j.matpur.2015.02.005.

[25]

P. D. Lax and X.-D. Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), 319-340.  doi: 10.1137/S1064827595291819.

[26]

J. Q. Li, T. Zhang and S. L. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, 98. Longman, Harlow, 1998.

[27]

J. Q. LiT. Zhang and Y. X. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations, Comm. Math. Phys., 267 (2006), 1-12.  doi: 10.1007/s00220-006-0033-1.

[28]

J. Q. Li and Y. X. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Rat. Mech. Anal., 193 (2009), 623-657.  doi: 10.1007/s00205-008-0140-6.

[29]

J. Q. Li and 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.  doi: 10.1007/s00220-010-1019-6.

[30]

M. J. Li and Y. X. Zheng, Semi-hyperbolic patches of solutions of the two-dimensional Euler equations, Arch. Rational Mech. Anal., 201 (2011), 1069-1096.  doi: 10.1007/s00205-011-0410-6.

[31]

C. S. Morawetz, On the non-existence of continuous transonic flow past profiles. Ⅱ, Comm. Pure Appl. Math., 10 (1957), 107-131.  doi: 10.1002/cpa.3160100105.

[32]

C. S. Morawetz, On a weak solution for a transonic flow problem, Comm. Pure Appl. Math., 38 (1985), 797-817.  doi: 10.1002/cpa.3160380610.

[33] O. A. Oleinik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form, Plenum Press, New York-London, 1973. 
[34]

C. W. Schulz-RinneJ. P. Collins and H. M. Glaz, Numerical solution of the Riemann problem for two-dimensional gsa dynamics, SIAM J. Sci. Compt., 14 (1993), 1394-1414.  doi: 10.1137/0914082.

[35]

W. C. Sheng and S. K. You, Interaction of a centered simple wave and a planar rarefaction wave of the two-dimensional Euler equations for pseudo-steady compressible flow, J. Math. Pures Appl., 114 (2018), 29-50.  doi: 10.1016/j.matpur.2017.07.019.

[36]

M. M. Smirnov, Equations of Mixed Type, Translations of Mathematical Monographs, 51. Amer. Math. Soc., Providence, R.I., 1978, 232 pp.

[37]

K. SongQ. Wang and Y. X. Zheng, The regularity of semihyperbolic patches near sonic lines for the 2-D Euler system in gas dynamics, SIAM J. Math. Anal., 47 (2015), 2200-2219.  doi: 10.1137/140964382.

[38]

K. Song and Y. X. Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system, Discrete Contin. Dyn. Syst., 24 (2009), 1365-1380.  doi: 10.3934/dcds.2009.24.1365.

[39]

C. P. Wang and Z. P. Xin, On a degenerate free boundary problem and continuous subsonic-sonic flows in a convergent nozzle, Arch. Ration. Mech. Anal., 208 (2013), 911-975.  doi: 10.1007/s00205-012-0607-3.

[40]

C. P. Wang and Z. P. Xin, Global smooth supersonic flows in infinite expanding nozzles, SIAM J. Math. Anal., 47 (2015), 3151-3211.  doi: 10.1137/140994289.

[41]

C. P. Wang and Z. P. Xin, Smooth transonic flows of Meyer type in de Laval nozzles, Arch. Ration. Mech. Anal., 232 (2019), 1597-1647.  doi: 10.1007/s00205-018-01350-9.

[42]

Q. Wang and Y. X. Zheng, The regularity of semi-hyperbolic patches at sonic lines for the pressure gradient equation in gas dynamics, Indiana Univ. Math. J., 63 (2014), 385-402.  doi: 10.1512/iumj.2014.63.5244.

[43]

C. J. Xie and Z. P. Xin, Global subsonic and subsonic-sonic flows through infinitely long nozzles, Indiana Univ. Math. J., 56 (2007), 2991-3023.  doi: 10.1512/iumj.2007.56.3108.

[44]

C. J. Xie and Z. P. Xin, Existence of global steady subsonic Euler flows through infinitely long nozzles, SIAM J. Math. Anal., 42 (2010), 751-784.  doi: 10.1137/09076667X.

[45]

Z. P. Xin and H. C. Yin, Transonic shock in a nozzle Ⅰ: Two-dimensional case, Comm. Pure Appl. Math., 58 (2005), 999-1050.  doi: 10.1002/cpa.20025.

[46]

T. Zhang and 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.  doi: 10.1137/0521032.

[47]

T. Y. Zhang and Y. X. Zheng, Sonic-supersonic solutions for the steady Euler equations, Indiana Univ. Math. J., 63 (2014), 1785-1817.  doi: 10.1512/iumj.2014.63.5434.

[48]

T. Y. Zhang and Y. X. Zheng, Existence of classical sonic-supersonic solutions for the pseudo steady Euler equations (in Chinese), Scientia Sinica Mathematica, 47 (2017), 1-18. 

[49]

Y. X. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems, Progress in Nonlinear Differential Equations and their Applications, 38. Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0141-0.

show all references

References:
[1]

L. Bers, On the continuation of a potential gas flow across the sonic line, Tech. Notes Nat. Adv. Comm. Aeronaut., 1950 (1950), 58 pp.

[2]

S. ČanićB. L. Keyfitz and E. H. Kim, A free boundary problem for a quasi-linear degenerate elliptic equation: Regular reflection of weak shocks, Comm. Pure Appl. Math., 55 (2002), 71-92.  doi: 10.1002/cpa.10013.

[3]

T. ChangG. Q. Chen and S. L. Yang, On the 2-D Riemann problem for the compressible Euler equations. Ⅰ. Interaction of shocks and rarefaction waves, Discrete Contin. Dyn. Syst., 1 (1995), 555-584.  doi: 10.3934/dcds.1995.1.555.

[4]

G.-Q. ChenC. M. DafermosM. Slemrod and D. H. Wang, On two-dimensional sonic-subsonic flow, Comm. Math. Phys., 271 (2007), 635-647.  doi: 10.1007/s00220-007-0211-9.

[5]

G.-Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc., 16 (2003), 461-494.  doi: 10.1090/S0894-0347-03-00422-3.

[6]

G.-Q. ChenF.-M. Huang and T.-Y. Wang, Subsonic-sonic limit of approximate solutions to multidimensional steady Euler equations, Arch. Ration. Mech. Anal., 219 (2016), 719-740.  doi: 10.1007/s00205-015-0905-7.

[7]

G.-Q. ChenM. Slemrod and D. H. Wang, Vanishing viscosity method for transonic flow, Arch. Ration. Mech. Anal., 189 (2008), 159-188.  doi: 10.1007/s00205-007-0101-5.

[8]

S. X. Chen, Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems, Trans. Amer. Math. Soc., 360 (2008), 5265-5289.  doi: 10.1090/S0002-9947-08-04493-0.

[9]

S. X. Chen, Compressible flow and transonic shock in a diverging nozzle, Comm. Math. Phys., 289 (2009), 75-106.  doi: 10.1007/s00220-009-0811-7.

[10]

S. X. Chen, The fundamental solution of the Keldysh type operator, Sci. China Ser. A, 52 (2009), 1829-1843.  doi: 10.1007/s11425-009-0069-8.

[11]

S. X. ChenZ. P. Xin and H. C. Yin, Global shock waves for the supersonic flow past a perturbed cone, Comm. Math. Phys., 228 (2002), 47-84.  doi: 10.1007/s002200200652.

[12]

X. Chen and Y. X. Zheng, The direct approach to the interaction of rarefaction waves of the two-dimensional Euler equations, Indiana Univ. Math. J., 59 (2010), 231-256.  doi: 10.1512/iumj.2010.59.3752.

[13]

L. P. Cook, Transonic Aerodynamics: Problems in Asymptotic Theory, Frontiers in Applied Mathematics, 12. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1993. doi: 10.1137/1.9781611970975.

[14]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y. 1948.

[15]

Z. H. Dai and T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Ration. Mech. Anal., 155 (2000), 277-298.  doi: 10.1007/s002050000113.

[16]

V. Elling and T.-P. Liu, Supersonic flow onto a solid wedge, Comm. Pure Appl. Math., 61 (2008), 1347-1448.  doi: 10.1002/cpa.20231.

[17]

C. Ferrari and F. Tricomi, Transonic Aerodynamics, Academic Press, 1968.

[18]

K. G. Guderley, The Theory of Transonic Flow, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962.

[19]

Y. B. Hu and J. Q. Li, Sonic-supersonic solutions for the two-dimensional steady full Euler equations, submitted, 2017.

[20]

Y. B. HuJ. Q. Li and W. C. Sheng, Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations, Z. Angew. Math. Phys., 63 (2012), 1021-1046.  doi: 10.1007/s00033-012-0203-2.

[21]

Y. B. Hu and T. Li, The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations, Comm. Pure Appl. Anal., 18 (2019), 3317-3336.  doi: 10.3934/cpaa.2019149.

[22]

Y. B. Hu and G. D. Wang, Semi-hyperbolic patches of solutions to the two-dimensional nonlinear wave system for Chaplygin gases, J. Differential Equations, 257 (2014), 1567-1590.  doi: 10.1016/j.jde.2014.05.020.

[23]

A. G. Kuz'min, Boundary Value Problems for Transonic Flow, John Wiley and Sons, 2002.

[24]

G. Lai and W. C. Sheng, Centered wave bubbles with sonic boundary of pseudosteady Guderley Mach reflection configurations in gas dynamics, J. Math. Pure Appl., 104 (2015), 179-206.  doi: 10.1016/j.matpur.2015.02.005.

[25]

P. D. Lax and X.-D. Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), 319-340.  doi: 10.1137/S1064827595291819.

[26]

J. Q. Li, T. Zhang and S. L. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, 98. Longman, Harlow, 1998.

[27]

J. Q. LiT. Zhang and Y. X. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations, Comm. Math. Phys., 267 (2006), 1-12.  doi: 10.1007/s00220-006-0033-1.

[28]

J. Q. Li and Y. X. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Rat. Mech. Anal., 193 (2009), 623-657.  doi: 10.1007/s00205-008-0140-6.

[29]

J. Q. Li and 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.  doi: 10.1007/s00220-010-1019-6.

[30]

M. J. Li and Y. X. Zheng, Semi-hyperbolic patches of solutions of the two-dimensional Euler equations, Arch. Rational Mech. Anal., 201 (2011), 1069-1096.  doi: 10.1007/s00205-011-0410-6.

[31]

C. S. Morawetz, On the non-existence of continuous transonic flow past profiles. Ⅱ, Comm. Pure Appl. Math., 10 (1957), 107-131.  doi: 10.1002/cpa.3160100105.

[32]

C. S. Morawetz, On a weak solution for a transonic flow problem, Comm. Pure Appl. Math., 38 (1985), 797-817.  doi: 10.1002/cpa.3160380610.

[33] O. A. Oleinik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form, Plenum Press, New York-London, 1973. 
[34]

C. W. Schulz-RinneJ. P. Collins and H. M. Glaz, Numerical solution of the Riemann problem for two-dimensional gsa dynamics, SIAM J. Sci. Compt., 14 (1993), 1394-1414.  doi: 10.1137/0914082.

[35]

W. C. Sheng and S. K. You, Interaction of a centered simple wave and a planar rarefaction wave of the two-dimensional Euler equations for pseudo-steady compressible flow, J. Math. Pures Appl., 114 (2018), 29-50.  doi: 10.1016/j.matpur.2017.07.019.

[36]

M. M. Smirnov, Equations of Mixed Type, Translations of Mathematical Monographs, 51. Amer. Math. Soc., Providence, R.I., 1978, 232 pp.

[37]

K. SongQ. Wang and Y. X. Zheng, The regularity of semihyperbolic patches near sonic lines for the 2-D Euler system in gas dynamics, SIAM J. Math. Anal., 47 (2015), 2200-2219.  doi: 10.1137/140964382.

[38]

K. Song and Y. X. Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system, Discrete Contin. Dyn. Syst., 24 (2009), 1365-1380.  doi: 10.3934/dcds.2009.24.1365.

[39]

C. P. Wang and Z. P. Xin, On a degenerate free boundary problem and continuous subsonic-sonic flows in a convergent nozzle, Arch. Ration. Mech. Anal., 208 (2013), 911-975.  doi: 10.1007/s00205-012-0607-3.

[40]

C. P. Wang and Z. P. Xin, Global smooth supersonic flows in infinite expanding nozzles, SIAM J. Math. Anal., 47 (2015), 3151-3211.  doi: 10.1137/140994289.

[41]

C. P. Wang and Z. P. Xin, Smooth transonic flows of Meyer type in de Laval nozzles, Arch. Ration. Mech. Anal., 232 (2019), 1597-1647.  doi: 10.1007/s00205-018-01350-9.

[42]

Q. Wang and Y. X. Zheng, The regularity of semi-hyperbolic patches at sonic lines for the pressure gradient equation in gas dynamics, Indiana Univ. Math. J., 63 (2014), 385-402.  doi: 10.1512/iumj.2014.63.5244.

[43]

C. J. Xie and Z. P. Xin, Global subsonic and subsonic-sonic flows through infinitely long nozzles, Indiana Univ. Math. J., 56 (2007), 2991-3023.  doi: 10.1512/iumj.2007.56.3108.

[44]

C. J. Xie and Z. P. Xin, Existence of global steady subsonic Euler flows through infinitely long nozzles, SIAM J. Math. Anal., 42 (2010), 751-784.  doi: 10.1137/09076667X.

[45]

Z. P. Xin and H. C. Yin, Transonic shock in a nozzle Ⅰ: Two-dimensional case, Comm. Pure Appl. Math., 58 (2005), 999-1050.  doi: 10.1002/cpa.20025.

[46]

T. Zhang and 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.  doi: 10.1137/0521032.

[47]

T. Y. Zhang and Y. X. Zheng, Sonic-supersonic solutions for the steady Euler equations, Indiana Univ. Math. J., 63 (2014), 1785-1817.  doi: 10.1512/iumj.2014.63.5434.

[48]

T. Y. Zhang and Y. X. Zheng, Existence of classical sonic-supersonic solutions for the pseudo steady Euler equations (in Chinese), Scientia Sinica Mathematica, 47 (2017), 1-18. 

[49]

Y. X. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems, Progress in Nonlinear Differential Equations and their Applications, 38. Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0141-0.

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