# 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. 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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

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##### 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
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