# American Institute of Mathematical Sciences

November  2014, 13(6): 2733-2748. doi: 10.3934/cpaa.2014.13.2733

## The two dimensional gas expansion problem of the Euler equations for the generalized Chaplygin gas

 1 Department of Mathematics, Shanghai University, Shanghai, 200444, China

Received  April 2014 Revised  June 2014 Published  July 2014

The collapse of a wedge-shaped dam containing fluid initially with a uniform velocity can be described as an expansion problem of gas into vacuum. It is an important class of binary interaction of rarefaction waves in the two dimensional Riemann problems for the compressible Euler equations. In this paper, we present various characteristic decompositions of the two dimensional pseudo-steady Euler equations for the generalized Chaplygin gas and obtain some priori estimates. By these estimates, we prove the global existence of solution to the expansion problem of a wedge of gas into vacuum with the half angle $\theta\in(0,\pi/2)$ for the generalized Chaplygin gas.
Citation: Ju Ge, Wancheng Sheng. The two dimensional gas expansion problem of the Euler equations for the generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2733-2748. doi: 10.3934/cpaa.2014.13.2733
##### References:
 [1] T. Chang, G. Q. Chen and S. Yang, On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves,, \emph{Discrete and Continuous Dynamical Systems}, 1 (1995), 555. doi: 10.3934/dcds.1995.1.555. Google Scholar [2] T. Chang, G. Q. Chen and S. Yang, On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities,, \emph{Discrete and Continuous Dynamical Systems}, 6 (2000), 419. doi: 10.3934/dcds.2000.6.419. Google Scholar [3] T. Chang and L. Hsiao, The Riemann problem and interaction of waves in gas dynamics,, \emph{Pitman Monographs, 41 (1989), 95. Google Scholar [4] S. X. Chen and A. F. Qu, Two-dimensional Riemann problems for Chaplygin gas,, \emph{SIAM J. Math. Anal.}, 44 (2012), 2146. doi: 10.1137/110838091. Google Scholar [5] X. Chen and Y. X. Zheng, The interaction of rarefaction waves of the two-dimensional Euler equations,, \emph{Indiana Univ. Math. J.}, 59 (2010), 231. doi: 10.1512/iumj.2010.59.3752. Google Scholar [6] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, New York, (1948). Google Scholar [7] R. Courant and D. Hilbert, Methodern der mathematischen Physik,, Vol.II. Springer, (1937). Google Scholar [8] Z. H. Dai and T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics,, \emph{Arch. Ration. Mech. Anal.}, 155 (2000), 277. doi: 10.1007/s002050000113. Google Scholar [9] L. H. Guo, W. C. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system,, \emph{Commun. Pure Appl. Anal.}, 9 (2010), 431. doi: 10.3934/cpaa.2010.9.431. Google Scholar [10] Y. B. Hu, J. Q. Li and W. C. Sheng, Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations,, \emph{Z. angew Math. Phys.}, 63 (2012), 1021. doi: 10.1007/s00033-012-0203-2. Google Scholar [11] Y. B. Hu and W. C. Sheng, Characteristic decomposition of the 2$\times$2 quasilinear strictly hyperbolic systems,, \emph{Appl. Math. Lett.}, 25 (2012), 262. doi: 10.1016/j.aml.2011.08.021. Google Scholar [12] D. X. Kong, K. F. Liu and Y. Z. Wang, Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases,, \emph{Sci. China Math.}, 53 (2010), 719. doi: 10.1007/s11425-010-0060-4. Google Scholar [13] G. Lai, W. C. Sheng and Y. X. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions,, \emph{Discrete Contin. Dyn. Syst.}, 31 (2011), 489. doi: 10.3934/dcds.2011.31.489. Google Scholar [14] L. E. Levine, The expansion of a wedge of gas into a vacuum,, \emph{Proc. Camb. Philol. Soc.}, 64 (1968), 1151. Google Scholar [15] J. Q. Li, On the two-dimensional gas expansion for compressible Euler equations,, \emph{SIAM J. Appl. Math.}, 62 (): 831. doi: 10.1137/S0036139900361349. Google Scholar [16] J. Q. Li, Global solution of an initial-value problem for two-dimesional compressible Euler equations,, \emph{J. Differential Equations}, 179 (2002), 178. doi: 10.1006/jdeq.2001.4025. Google Scholar [17] J. Q. Li, Z. C. Yang and Y. X. Zheng, Characteristic decompositions and interactions of rarefaction waves of 2-d Euler equations,, \emph{J. Differential Equations}, 250 (2011), 782. doi: 10.1016/j.jde.2010.07.009. Google Scholar [18] J. Q. Li, T. Zhang and Y. X. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations,, \emph{Commu. Math. Phys.}, 267 (2006), 1. doi: 10.1007/s00220-006-0033-1. Google Scholar [19] J. Q. Li and Y. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations,, \emph{Arch. Rational. Mech. Anal.}, 193 (2009), 623. doi: 10.1007/s00205-008-0140-6. Google Scholar [20] J. Q. Li and Y. X. Zheng, Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations,, \emph{Commun. Math. Phys.}, 296 (2010), 303. doi: 10.1007/s00220-010-1019-6. Google Scholar [21] T. T. Li and W. C. Yu, Boundary value problems of quasilinear hyperbolic system,, Duke University Mathematics Series V, (1985). Google Scholar [22] V. V. Meleshko and G. J. F. van Heijst, On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid,, \emph{J. Fluid Mech.}, 272 (1994), 157. doi: 10.1017/S0022112094004428. Google Scholar [23] D. Serre, Multidimensional shock interaction for a Chaplygin gas,, \emph{Arch. Rational Mech. Anal.}, 191 (2009), 539. doi: 10.1007/s00205-008-0110-z. Google Scholar [24] V. A. Suchkov, Flow into a vacuum along an oblique wall,, \emph{J. Appl. Math. Mech.}, 27 (1963), 1132. Google Scholar [25] G. D. Wang, B. C. Chen and Y. B. Hu, The two-dimensional Riemann problem for Chaplygin gas dynamics with three constant states,, \emph{J. Math. Anal. Appl.}, 393 (2012), 544. doi: 10.1016/j.jmaa.2012.03.017. Google Scholar [26] R. H. Wang and Z. Q. Wu, On mixed initial boundary value problem for quasi-linear hyperbolic system of partial differential equations in two independent variables (in Chinese),, \emph{Acta. Sci. Nat. Jilin. Univ.}, 2 (1963), 459. Google Scholar [27] T. Zhang and Y. X. Zheng, Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems,, \emph{SIAM J. Math. Anal.}, 21 (1990), 593. doi: 10.1137/0521032. Google Scholar [28] W. X. Zhao, The expansion of gas from a wedge with small angle into a vaccum,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 2319. doi: 10.3934/cpaa.2013.12.2319. Google Scholar [29] Y. Zheng, The compressible Euler system in two space dimensions,, Nonlinear conservation laws, (2009), 301. doi: 10.1142/9789814273282_0005. Google Scholar

show all references

##### References:
 [1] T. Chang, G. Q. Chen and S. Yang, On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves,, \emph{Discrete and Continuous Dynamical Systems}, 1 (1995), 555. doi: 10.3934/dcds.1995.1.555. Google Scholar [2] T. Chang, G. Q. Chen and S. Yang, On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities,, \emph{Discrete and Continuous Dynamical Systems}, 6 (2000), 419. doi: 10.3934/dcds.2000.6.419. Google Scholar [3] T. Chang and L. Hsiao, The Riemann problem and interaction of waves in gas dynamics,, \emph{Pitman Monographs, 41 (1989), 95. Google Scholar [4] S. X. Chen and A. F. Qu, Two-dimensional Riemann problems for Chaplygin gas,, \emph{SIAM J. Math. Anal.}, 44 (2012), 2146. doi: 10.1137/110838091. Google Scholar [5] X. Chen and Y. X. Zheng, The interaction of rarefaction waves of the two-dimensional Euler equations,, \emph{Indiana Univ. Math. J.}, 59 (2010), 231. doi: 10.1512/iumj.2010.59.3752. Google Scholar [6] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, New York, (1948). Google Scholar [7] R. Courant and D. Hilbert, Methodern der mathematischen Physik,, Vol.II. Springer, (1937). Google Scholar [8] Z. H. Dai and T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics,, \emph{Arch. Ration. Mech. Anal.}, 155 (2000), 277. doi: 10.1007/s002050000113. Google Scholar [9] L. H. Guo, W. C. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system,, \emph{Commun. Pure Appl. Anal.}, 9 (2010), 431. doi: 10.3934/cpaa.2010.9.431. Google Scholar [10] Y. B. Hu, J. Q. Li and W. C. Sheng, Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations,, \emph{Z. angew Math. Phys.}, 63 (2012), 1021. doi: 10.1007/s00033-012-0203-2. Google Scholar [11] Y. B. Hu and W. C. Sheng, Characteristic decomposition of the 2$\times$2 quasilinear strictly hyperbolic systems,, \emph{Appl. Math. Lett.}, 25 (2012), 262. doi: 10.1016/j.aml.2011.08.021. Google Scholar [12] D. X. Kong, K. F. Liu and Y. Z. Wang, Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases,, \emph{Sci. China Math.}, 53 (2010), 719. doi: 10.1007/s11425-010-0060-4. Google Scholar [13] G. Lai, W. C. Sheng and Y. X. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions,, \emph{Discrete Contin. Dyn. Syst.}, 31 (2011), 489. doi: 10.3934/dcds.2011.31.489. Google Scholar [14] L. E. Levine, The expansion of a wedge of gas into a vacuum,, \emph{Proc. Camb. Philol. Soc.}, 64 (1968), 1151. Google Scholar [15] J. Q. Li, On the two-dimensional gas expansion for compressible Euler equations,, \emph{SIAM J. Appl. Math.}, 62 (): 831. doi: 10.1137/S0036139900361349. Google Scholar [16] J. Q. Li, Global solution of an initial-value problem for two-dimesional compressible Euler equations,, \emph{J. Differential Equations}, 179 (2002), 178. doi: 10.1006/jdeq.2001.4025. Google Scholar [17] J. Q. Li, Z. C. Yang and Y. X. Zheng, Characteristic decompositions and interactions of rarefaction waves of 2-d Euler equations,, \emph{J. Differential Equations}, 250 (2011), 782. doi: 10.1016/j.jde.2010.07.009. Google Scholar [18] J. Q. Li, T. Zhang and Y. X. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations,, \emph{Commu. Math. Phys.}, 267 (2006), 1. doi: 10.1007/s00220-006-0033-1. Google Scholar [19] J. Q. Li and Y. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations,, \emph{Arch. Rational. Mech. Anal.}, 193 (2009), 623. doi: 10.1007/s00205-008-0140-6. Google Scholar [20] J. Q. Li and Y. X. Zheng, Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations,, \emph{Commun. Math. Phys.}, 296 (2010), 303. doi: 10.1007/s00220-010-1019-6. Google Scholar [21] T. T. Li and W. C. Yu, Boundary value problems of quasilinear hyperbolic system,, Duke University Mathematics Series V, (1985). Google Scholar [22] V. V. Meleshko and G. J. F. van Heijst, On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid,, \emph{J. Fluid Mech.}, 272 (1994), 157. doi: 10.1017/S0022112094004428. Google Scholar [23] D. Serre, Multidimensional shock interaction for a Chaplygin gas,, \emph{Arch. Rational Mech. Anal.}, 191 (2009), 539. doi: 10.1007/s00205-008-0110-z. Google Scholar [24] V. A. Suchkov, Flow into a vacuum along an oblique wall,, \emph{J. Appl. Math. Mech.}, 27 (1963), 1132. Google Scholar [25] G. D. Wang, B. C. Chen and Y. B. Hu, The two-dimensional Riemann problem for Chaplygin gas dynamics with three constant states,, \emph{J. Math. Anal. Appl.}, 393 (2012), 544. doi: 10.1016/j.jmaa.2012.03.017. Google Scholar [26] R. H. Wang and Z. Q. Wu, On mixed initial boundary value problem for quasi-linear hyperbolic system of partial differential equations in two independent variables (in Chinese),, \emph{Acta. Sci. Nat. Jilin. Univ.}, 2 (1963), 459. Google Scholar [27] T. Zhang and Y. X. Zheng, Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems,, \emph{SIAM J. Math. Anal.}, 21 (1990), 593. doi: 10.1137/0521032. Google Scholar [28] W. X. Zhao, The expansion of gas from a wedge with small angle into a vaccum,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 2319. doi: 10.3934/cpaa.2013.12.2319. Google Scholar [29] Y. Zheng, The compressible Euler system in two space dimensions,, Nonlinear conservation laws, (2009), 301. doi: 10.1142/9789814273282_0005. Google Scholar
 [1] Jianjun Chen, Wancheng Sheng. The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2018, 17 (1) : 127-142. doi: 10.3934/cpaa.2018008 [2] Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127 [3] Lihui Guo, Wancheng Sheng, Tong Zhang. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system$^*$. Communications on Pure & Applied Analysis, 2010, 9 (2) : 431-458. doi: 10.3934/cpaa.2010.9.431 [4] Qin Wang, Kyungwoo Song. The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1661-1675. doi: 10.3934/dcds.2016.36.1661 [5] Marko Nedeljkov, Sanja Ružičić. On the uniqueness of solution to generalized Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4439-4460. doi: 10.3934/dcds.2017190 [6] Lihui Guo, Tong Li, Gan Yin. The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. Communications on Pure & Applied Analysis, 2017, 16 (1) : 295-310. doi: 10.3934/cpaa.2017014 [7] Yanbo Hu, Tong Li. Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations. Kinetic & Related Models, 2019, 12 (6) : 1197-1228. doi: 10.3934/krm.2019046 [8] Eun Heui Kim, Charis Tsikkou. Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6257-6289. doi: 10.3934/dcds.2017271 [9] Huahui Li, Zhiqiang Shao. Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2373-2400. doi: 10.3934/cpaa.2016041 [10] 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 & Continuous Dynamical Systems - A, 1995, 1 (4) : 555-584. doi: 10.3934/dcds.1995.1.555 [11] Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985 [12] Mapundi K. Banda, Michael Herty, Axel Klar. Gas flow in pipeline networks. Networks & Heterogeneous Media, 2006, 1 (1) : 41-56. doi: 10.3934/nhm.2006.1.41 [13] Geng Lai, Wancheng Sheng, Yuxi Zheng. Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 489-523. doi: 10.3934/dcds.2011.31.489 [14] Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687 [15] Hisashi Okamoto, Takashi Sakajo, Marcus Wunsch. Steady-states and traveling-wave solutions of the generalized Constantin--Lax--Majda equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3155-3170. doi: 10.3934/dcds.2014.34.3155 [16] W. G. Litvinov. Problem on stationary flow of electrorheological fluids at the generalized conditions of slip on the boundary. Communications on Pure & Applied Analysis, 2007, 6 (1) : 247-277. doi: 10.3934/cpaa.2007.6.247 [17] Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759 [18] Feimin Huang, Yi Wang, Tong Yang. Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinetic & Related Models, 2010, 3 (4) : 685-728. doi: 10.3934/krm.2010.3.685 [19] Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure & Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755 [20] Alberto Bressan, Fang Yu. Continuous Riemann solvers for traffic flow at a junction. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4149-4171. doi: 10.3934/dcds.2015.35.4149

2018 Impact Factor: 0.925