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November  2019, 18(6): 3317-3336. doi: 10.3934/cpaa.2019149

The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations

Yanbo Hu 1,, and  Tong Li 2,

1. 

Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China
2. 

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

* Corresponding author

Received  August 2018 Revised  February 2019 Published  May 2019

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

We study the regularity of solution and of sonic boundary to a degenerate Goursat problem originated from the two-dimensional Riemann problem of the compressible isothermal Euler equations. By using the ideas of characteristic decomposition and the bootstrap method, we show that the solution is uniformly ${C^{1,\frac{1}{6}}}$ up to the degenerate sonic boundary and that the sonic curve is ${C^{1,\frac{1}{6}}}$.

Keywords: Compressible Euler equations, semi-hyperbolic patch, degenerate Goursat problem, sonic curve, characteristic decomposition.
Mathematics Subject Classification: 35L65, 35L80, 35R35.
Citation: Yanbo Hu, Tong Li. The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3317-3336. doi: 10.3934/cpaa.2019149
References:
[1]

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.  Google Scholar

[2]

J. D. Cole and L. P. Cook, Transonic Aerodynamics, North-Holland Series in Applied Mathematics and Mechanics, 1986. doi: 10.1137/1.9781611970975.  Google Scholar

[3]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948.  Google Scholar

[4]

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.  Google Scholar

[5]

G. Glimm, X. Ji, J. Li, X. Li, P. Zhang, T. Zhang and Y. Zheng, Transonic shock formation in a rarefaction Riemann problem for the 2-D compressible Euler equations, SIAM J. Appl. Math., 69 (2008), 720-742.  doi: 10.1137/07070632X.  Google Scholar

[6]

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

[7]

Y. B. Hu, J. 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.  Google Scholar

[8]

Y. B. Hu and T. Li, An improved regularity result of semi-hyperbolic patch problems for the 2-D isentropic Euler equations, J. Math. Anal. Appl., 467 (2018), 1174-1193.  doi: 10.1016/j.jmaa.2018.07.064.  Google Scholar

[9]

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), 1579-1590.  doi: 10.1016/j.jde.2014.05.020.  Google Scholar

[10]

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.  Google Scholar

[11]

J. Q. Li, Z. C. Yang and Y. X. Zheng, Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations, J. Differential Equations, 250 (2011), 782-798.  doi: 10.1016/j.jde.2010.07.009.  Google Scholar

[12]

J. Q. Li, T. Zhang and S. L. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Longman, Harlow, 1998.  Google Scholar

[13]

J. Q. Li, T. 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.  Google Scholar

[14]

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.  Google Scholar

[15]

J. Q. Li and Y. 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.  Google Scholar

[16]

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.  Google Scholar

[17]

W. C. Sheng, G. D. Wang and 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.  doi: 10.1137/090760362.  Google Scholar

[18]

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.  Google Scholar

[19]

K. Song, Semi-hyperbolic patches arising from a transonic shock in simple waves interaction, J. Korean Math. Soc., 50 (2013), 945-957.  doi: 10.4134/JKMS.2013.50.5.945.  Google Scholar

[20]

K. Song, Q. 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.  Google Scholar

[21]

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

[22]

A. M. Tesdall, R. Sanders and B. L. Keyfitz, The triple point paradox for the nonlinear wave system, SIAM J. Appl. Math., 67 (2006), 321-336.  doi: 10.1137/060660758.  Google Scholar

[23]

A. M. Tesdall, R. Sanders and B. L. Keyfitz, Self-similar solutions for the triple point paradox in gasdynamics, SIAM J. Appl. Math., 68 (2008), 1360-1377.  doi: 10.1137/070698567.  Google Scholar

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

T. Y. Zhang and Y. X. Zheng, The structure of solutions near a sonic line in gas dynamics via the pressure gradient equation, J. Math. Anal. Appl., 443 (2016), 39-56.  doi: 10.1016/j.jmaa.2016.04.002.  Google Scholar

[27]

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.  doi: 10.1512/iumj.2014.63.5434.  Google Scholar

[28]

Y. X. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems, Birkhauser, Boston, 2001. doi: 10.1007/978-1-4612-0141-0.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

J. D. Cole and L. P. Cook, Transonic Aerodynamics, North-Holland Series in Applied Mathematics and Mechanics, 1986. doi: 10.1137/1.9781611970975.  Google Scholar

[3]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948.  Google Scholar

[4]

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.  Google Scholar

[5]

G. Glimm, X. Ji, J. Li, X. Li, P. Zhang, T. Zhang and Y. Zheng, Transonic shock formation in a rarefaction Riemann problem for the 2-D compressible Euler equations, SIAM J. Appl. Math., 69 (2008), 720-742.  doi: 10.1137/07070632X.  Google Scholar

[6]

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

[7]

Y. B. Hu, J. 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.  Google Scholar

[8]

Y. B. Hu and T. Li, An improved regularity result of semi-hyperbolic patch problems for the 2-D isentropic Euler equations, J. Math. Anal. Appl., 467 (2018), 1174-1193.  doi: 10.1016/j.jmaa.2018.07.064.  Google Scholar

[9]

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), 1579-1590.  doi: 10.1016/j.jde.2014.05.020.  Google Scholar

[10]

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.  Google Scholar

[11]

J. Q. Li, Z. C. Yang and Y. X. Zheng, Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations, J. Differential Equations, 250 (2011), 782-798.  doi: 10.1016/j.jde.2010.07.009.  Google Scholar

[12]

J. Q. Li, T. Zhang and S. L. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Longman, Harlow, 1998.  Google Scholar

[13]

J. Q. Li, T. 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.  Google Scholar

[14]

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.  Google Scholar

[15]

J. Q. Li and Y. 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.  Google Scholar

[16]

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.  Google Scholar

[17]

W. C. Sheng, G. D. Wang and 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.  doi: 10.1137/090760362.  Google Scholar

[18]

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.  Google Scholar

[19]

K. Song, Semi-hyperbolic patches arising from a transonic shock in simple waves interaction, J. Korean Math. Soc., 50 (2013), 945-957.  doi: 10.4134/JKMS.2013.50.5.945.  Google Scholar

[20]

K. Song, Q. 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.  Google Scholar

[21]

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

[22]

A. M. Tesdall, R. Sanders and B. L. Keyfitz, The triple point paradox for the nonlinear wave system, SIAM J. Appl. Math., 67 (2006), 321-336.  doi: 10.1137/060660758.  Google Scholar

[23]

A. M. Tesdall, R. Sanders and B. L. Keyfitz, Self-similar solutions for the triple point paradox in gasdynamics, SIAM J. Appl. Math., 68 (2008), 1360-1377.  doi: 10.1137/070698567.  Google Scholar

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

T. Y. Zhang and Y. X. Zheng, The structure of solutions near a sonic line in gas dynamics via the pressure gradient equation, J. Math. Anal. Appl., 443 (2016), 39-56.  doi: 10.1016/j.jmaa.2016.04.002.  Google Scholar

[27]

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.  doi: 10.1512/iumj.2014.63.5434.  Google Scholar

[28]

Y. X. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems, Birkhauser, Boston, 2001. doi: 10.1007/978-1-4612-0141-0.  Google Scholar

Figure 1.  The semi-hyperbolic patch
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Figure 2.  Case 2
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Figure 3.  The region of $ \Omega_\nu(\bar{z}) $
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