
-
Previous Article
Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term
- CPAA Home
- This Issue
-
Next Article
The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line
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 ${C^{1,\frac{1}{6}}}$.
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. |
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. |



[1] |
Jianjun Chen, Geng Lai. Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations. Communications on Pure and Applied Analysis, 2019, 18 (2) : 943-958. doi: 10.3934/cpaa.2019046 |
[2] |
Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605 |
[3] |
Yanbo Hu, Jiequan Li. On a supersonic-sonic patch arising from the frankl problem in transonic flows. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2643-2663. doi: 10.3934/cpaa.2021015 |
[4] |
Kyungwoo Song, Yuxi Zheng. Semi-hyperbolic patches of solutions of the pressure gradient system. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1365-1380. doi: 10.3934/dcds.2009.24.1365 |
[5] |
Hiroki Sumi, Mariusz Urbański. Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 313-363. doi: 10.3934/dcds.2011.30.313 |
[6] |
Shuxing Chen, Gui-Qiang Chen, Zejun Wang, Dehua Wang. A multidimensional piston problem for the Euler equations for compressible flow. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 361-383. doi: 10.3934/dcds.2005.13.361 |
[7] |
Guanming Gai, Yuanyuan Nie, Chunpeng Wang. A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2555-2577. doi: 10.3934/cpaa.2021070 |
[8] |
Magdalena Caubergh, Freddy Dumortier, Stijn Luca. Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Lienard systems. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 963-980. doi: 10.3934/dcds.2010.27.963 |
[9] |
Zhi-Qiang Shao. Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2739-2752. doi: 10.3934/cpaa.2013.12.2739 |
[10] |
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 |
[11] |
Gabriele Beltramo, Primoz Skraba, Rayna Andreeva, Rik Sarkar, Ylenia Giarratano, Miguel O. Bernabeu. Euler characteristic surfaces. Foundations of Data Science, 2021 doi: 10.3934/fods.2021027 |
[12] |
Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic and Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687 |
[13] |
Chengchun Hao. Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2885-2931. doi: 10.3934/dcdsb.2015.20.2885 |
[14] |
Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 555-584. doi: 10.3934/dcds.1995.1.555 |
[15] |
Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 419-430. doi: 10.3934/dcds.2000.6.419 |
[16] |
Ping Chen, Ting Zhang. A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients. Communications on Pure and Applied Analysis, 2008, 7 (4) : 987-1016. doi: 10.3934/cpaa.2008.7.987 |
[17] |
Young-Pil Choi. Compressible Euler equations interacting with incompressible flow. Kinetic and Related Models, 2015, 8 (2) : 335-358. doi: 10.3934/krm.2015.8.335 |
[18] |
Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic and Related Models, 2014, 7 (4) : 605-619. doi: 10.3934/krm.2014.7.605 |
[19] |
Jianwei Yang, Dongling Li, Xiao Yang. On the quasineutral limit for the compressible Euler-Poisson equations. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022020 |
[20] |
Quentin Chauleur. The isothermal limit for the compressible Euler equations with damping. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022059 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]