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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

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 ${C^{1,\frac{1}{6}}}$.

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. GlimmX. JiJ. LiX. LiP. ZhangT. 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. 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. 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. LiZ. 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. 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. 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. ShengG. 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. 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. 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. TesdallR. 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. TesdallR. 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. GlimmX. JiJ. LiX. LiP. ZhangT. 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. 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. 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. LiZ. 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. 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. 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. ShengG. 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. 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. 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. TesdallR. 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. TesdallR. 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
Figure 2.  Case 2
Figure 3.  The region of $ \Omega_\nu(\bar{z}) $
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