Figure 1.
The semi-hyperbolic patch
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The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line
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 |
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 and Applied Analysis,
2019, 18
(6)
: 3317-3336.
doi: 10.3934/cpaa.2019149
![]()
References:
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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. |
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J. D. Cole and L. P. Cook, Transonic Aerodynamics, North-Holland Series in Applied Mathematics and Mechanics, 1986.
doi: 10.1137/1.9781611970975. |
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R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948. |
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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. |
| [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. |
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Y. B. Hu and J. Q. Li, Sonic-supersonic solutions for the two-dimensional steady full Euler equations, submitted, 2017. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
J. Q. Li, T. Zhang and S. L. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Longman, Harlow, 1998. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [28] |
Y. X. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems, Birkhauser, Boston, 2001.
doi: 10.1007/978-1-4612-0141-0. |
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. |
| [2] |
J. D. Cole and L. P. Cook, Transonic Aerodynamics, North-Holland Series in Applied Mathematics and Mechanics, 1986.
doi: 10.1137/1.9781611970975. |
| [3] |
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948. |
| [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. |
| [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. |
| [6] |
Y. B. Hu and J. Q. Li, Sonic-supersonic solutions for the two-dimensional steady full Euler equations, submitted, 2017. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
J. Q. Li, T. Zhang and S. L. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Longman, Harlow, 1998. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [28] |
Y. X. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems, Birkhauser, Boston, 2001.
doi: 10.1007/978-1-4612-0141-0. |
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