# 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 and 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, Discrete and Continuous Dynamical Systems, 1 (1995), 555-584. doi: 10.3934/dcds.1995.1.555. [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, Discrete and Continuous Dynamical Systems, 6 (2000), 419-430. doi: 10.3934/dcds.2000.6.419. [3] T. Chang and L. Hsiao, The Riemann problem and interaction of waves in gas dynamics, Pitman Monographs, Longman Scientific and technica, 41 (1989), 95-161. [4] S. X. Chen and A. F. Qu, Two-dimensional Riemann problems for Chaplygin gas, SIAM J. Math. Anal., 44 (2012), 2146-2178. doi: 10.1137/110838091. [5] X. Chen and Y. X. Zheng, 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. [6] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, New York, Interscience, 1948. [7] R. Courant and D. Hilbert, Methodern der mathematischen Physik, Vol.II. Springer, Berlin, 1937. (Reprint: Interscience, New York, 1943.) [8] 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. [9] L. H. Guo, W. C. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal., 9 (2010), 431-458. doi: 10.3934/cpaa.2010.9.431. [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, Z. angew Math. Phys., 63 (2012), 1021-1046. doi: 10.1007/s00033-012-0203-2. [11] Y. B. Hu and W. C. Sheng, Characteristic decomposition of the 2$\times$2 quasilinear strictly hyperbolic systems, Appl. Math. Lett., 25 (2012), 262-267. doi: 10.1016/j.aml.2011.08.021. [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, Sci. China Math., 53 (2010), 719-738. doi: 10.1007/s11425-010-0060-4. [13] G. Lai, W. C. Sheng and Y. X. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions, Discrete Contin. Dyn. Syst., 31 (2011), 489-523. doi: 10.3934/dcds.2011.31.489. [14] L. E. Levine, The expansion of a wedge of gas into a vacuum, Proc. Camb. Philol. Soc., 64 (1968), 1151-1163. [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. [16] J. Q. Li, Global solution of an initial-value problem for two-dimesional compressible Euler equations, J. Differential Equations, 179 (2002), 178-194. doi: 10.1006/jdeq.2001.4025. [17] 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. [18] J. Q. Li, T. Zhang and Y. X. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations, Commu. Math. Phys., 267 (2006), 1-12. doi: 10.1007/s00220-006-0033-1. [19] J. Q. Li and Y. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Rational. Mech. Anal., 193 (2009), 623-657. doi: 10.1007/s00205-008-0140-6. [20] J. Q. Li and Y. X. Zheng, Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations, Commun. Math. Phys., 296 (2010), 303-321. doi: 10.1007/s00220-010-1019-6. [21] T. T. Li and W. C. Yu, Boundary value problems of quasilinear hyperbolic system, Duke University Mathematics Series V, Durham, 1985. [22] V. V. Meleshko and G. J. F. van Heijst, On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid, J. Fluid Mech., 272 (1994), 157-182. doi: 10.1017/S0022112094004428. [23] D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Rational Mech. Anal., 191 (2009), 539-577. doi: 10.1007/s00205-008-0110-z. [24] V. A. Suchkov, Flow into a vacuum along an oblique wall, J. Appl. Math. Mech., 27 (1963), 1132-1134. [25] G. D. Wang, B. C. Chen and Y. B. Hu, The two-dimensional Riemann problem for Chaplygin gas dynamics with three constant states, J. Math. Anal. Appl., 393 (2012), 544-562. doi: 10.1016/j.jmaa.2012.03.017. [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), Acta. Sci. Nat. Jilin. Univ., 2 (1963), 459-502. [27] T. Zhang and Y. X. Zheng, Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Math. Anal., 21 (1990), 593-630. doi: 10.1137/0521032. [28] W. X. Zhao, The expansion of gas from a wedge with small angle into a vaccum, Comm. Pure Appl. Anal., 12 (2013), 2319-2330. doi: 10.3934/cpaa.2013.12.2319. [29] Y. Zheng, The compressible Euler system in two space dimensions, Nonlinear conservation laws, fluid systems and related topics, 301-390, Ser. Contemp. Appl. Math. CAM, 13, World Sci. Publishing, Singapore, 2009. doi: 10.1142/9789814273282_0005.

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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, Discrete and Continuous Dynamical Systems, 1 (1995), 555-584. doi: 10.3934/dcds.1995.1.555. [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, Discrete and Continuous Dynamical Systems, 6 (2000), 419-430. doi: 10.3934/dcds.2000.6.419. [3] T. Chang and L. Hsiao, The Riemann problem and interaction of waves in gas dynamics, Pitman Monographs, Longman Scientific and technica, 41 (1989), 95-161. [4] S. X. Chen and A. F. Qu, Two-dimensional Riemann problems for Chaplygin gas, SIAM J. Math. Anal., 44 (2012), 2146-2178. doi: 10.1137/110838091. [5] X. Chen and Y. X. Zheng, 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. [6] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, New York, Interscience, 1948. [7] R. Courant and D. Hilbert, Methodern der mathematischen Physik, Vol.II. Springer, Berlin, 1937. (Reprint: Interscience, New York, 1943.) [8] 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. [9] L. H. Guo, W. C. Sheng and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal., 9 (2010), 431-458. doi: 10.3934/cpaa.2010.9.431. [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, Z. angew Math. Phys., 63 (2012), 1021-1046. doi: 10.1007/s00033-012-0203-2. [11] Y. B. Hu and W. C. Sheng, Characteristic decomposition of the 2$\times$2 quasilinear strictly hyperbolic systems, Appl. Math. Lett., 25 (2012), 262-267. doi: 10.1016/j.aml.2011.08.021. [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, Sci. China Math., 53 (2010), 719-738. doi: 10.1007/s11425-010-0060-4. [13] G. Lai, W. C. Sheng and Y. X. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions, Discrete Contin. Dyn. Syst., 31 (2011), 489-523. doi: 10.3934/dcds.2011.31.489. [14] L. E. Levine, The expansion of a wedge of gas into a vacuum, Proc. Camb. Philol. Soc., 64 (1968), 1151-1163. [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. [16] J. Q. Li, Global solution of an initial-value problem for two-dimesional compressible Euler equations, J. Differential Equations, 179 (2002), 178-194. doi: 10.1006/jdeq.2001.4025. [17] 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. [18] J. Q. Li, T. Zhang and Y. X. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations, Commu. Math. Phys., 267 (2006), 1-12. doi: 10.1007/s00220-006-0033-1. [19] J. Q. Li and Y. Zheng, Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Rational. Mech. Anal., 193 (2009), 623-657. doi: 10.1007/s00205-008-0140-6. [20] J. Q. Li and Y. X. Zheng, Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations, Commun. Math. Phys., 296 (2010), 303-321. doi: 10.1007/s00220-010-1019-6. [21] T. T. Li and W. C. Yu, Boundary value problems of quasilinear hyperbolic system, Duke University Mathematics Series V, Durham, 1985. [22] V. V. Meleshko and G. J. F. van Heijst, On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid, J. Fluid Mech., 272 (1994), 157-182. doi: 10.1017/S0022112094004428. [23] D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Rational Mech. Anal., 191 (2009), 539-577. doi: 10.1007/s00205-008-0110-z. [24] V. A. Suchkov, Flow into a vacuum along an oblique wall, J. Appl. Math. Mech., 27 (1963), 1132-1134. [25] G. D. Wang, B. C. Chen and Y. B. Hu, The two-dimensional Riemann problem for Chaplygin gas dynamics with three constant states, J. Math. Anal. Appl., 393 (2012), 544-562. doi: 10.1016/j.jmaa.2012.03.017. [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), Acta. Sci. Nat. Jilin. Univ., 2 (1963), 459-502. [27] T. Zhang and Y. X. Zheng, Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Math. Anal., 21 (1990), 593-630. doi: 10.1137/0521032. [28] W. X. Zhao, The expansion of gas from a wedge with small angle into a vaccum, Comm. Pure Appl. Anal., 12 (2013), 2319-2330. doi: 10.3934/cpaa.2013.12.2319. [29] Y. Zheng, The compressible Euler system in two space dimensions, Nonlinear conservation laws, fluid systems and related topics, 301-390, Ser. Contemp. Appl. Math. CAM, 13, World Sci. Publishing, Singapore, 2009. doi: 10.1142/9789814273282_0005.
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