# American Institute of Mathematical Sciences

June  2011, 31(2): 489-523. doi: 10.3934/dcds.2011.31.489

## Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions

 1 Department of Mathematics, Shanghai University, Shanghai 200444, China, China 2 Department of Mathematical Sciences, Yeshiva University, New York, NY 10033, United States

Received  February 2010 Revised  March 2011 Published  June 2011

We present two new types of self-similar solutions to the Chaplygin gas model in two space dimensions: Simple waves and pressure delta waves, which are absent in one space dimension, but appear in the solutions to the two-dimensional Riemann problems. A simple wave is a flow in a physical region whose image in the state space is a one-dimensional curve. The solutions to the interaction of two rarefaction simple waves are constructed. Comparisons with polytropic gases are made. Pressure delta waves are Dirac type concentration in the pressure variable, or impulses of the pressure on discontinuities. They appear in the study of Riemann problems of four rarefaction shocks. This type of discontinuities and concentrations are different from delta waves for the pressureless gas flow model, for which the delta waves are associated with convection and concentration of mass. By re-interpreting the terms in the Chaplygin gas system into new forms we are able to define distributional solutions that include the pressure delta waves. Generalized Rankine-Hugoniot conditions for pressure delta waves are derived.
Citation: Geng Lai, Wancheng Sheng, Yuxi Zheng. Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 489-523. doi: 10.3934/dcds.2011.31.489
##### References:
 [1] S. Bang, Interaction of three and four rarefaction waves of the pressure-gradient system, J. Differential Equations, 246 (2009), 453-481. [2] Y. Brenier, Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations, J. Math. Fluid Mech., 7 (2005), 326-331. doi: 10.1007/s00021-005-0162-x. [3] S. X. Chen and A. F. Qu, Interaction of rarefaction waves in jet stream, J. Differential Equations, 248 (2010), 2931-2954. [4] G. Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanish pressure limit of solutions to the Euler equations for isentropic fliuds, SIAM J. Math. Anal., 34 (2003), 925-938. doi: 10.1137/S0036141001399350. [5] R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Interscience, New York, 1948. [6] Z. Dai ang T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Rat. Mech. Anal., 155 (2000), 277-298. doi: 10.1007/s002050000113. [7] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, Springer-Verlag, 2003. [8] L. H. Guo, W. C. Sheng and T. Zhang, The 2D Riemann problem for isentropic Chaplygin gas dynamic system, Comm. Pure Appl. Anal., 9 (2010), 431-458. doi: 10.3934/cpaa.2010.9.431. [9] Y. B. Hu, J. Q. Li and W. C. Sheng, Interaction of rarefaction waves for 2D isothermal Euler equations, submitted for publication, 2011. [10] F. Huang and Z. Wang, Well posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146. doi: 10.1007/s002200100506. [11] F. John, "Partial Differential Equations," Springer-Verlag, 1982. [12] B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations, 118 (1995), 420-451. [13] D. J. Korchinski, "Solutions of a Riemann Problem for A 2 $\times$ 2 System of Conservation Laws Prosssing No Classical Solutions," thesis, Adelphi University, Garden City, NY, 1977. [14] N. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation, In "Nonlinear Functional Analysis and its Applications," Proc. Symp. Pure Math., 45 (1986), 81-90, Providence, Amer. Math. Soc.. [15] G. Lai and W. C. Sheng, Simple waves for 2D isentropic ir-rotational self-similar Euler system, Appl. Math. Mech, 31 (2010), 1-12. [16] P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. [17] Z. Lei and Y. Zheng, A complete global solution to the pressure gradient equation, J. Differential Equations, 236 (2007), 280-292. [18] J. Li, On the 2D gas expansion for compressible Euler euqations, SIAM J. Appl. Math., 62 (2001), 831-852. doi: 10.1137/S0036139900361349. [19] J. Li, Global solution of an initial-value problem for 2D compressible Euler equations, J. Differential Equations, 179 (2002), 178-194. [20] J. Li and H. Yang, Delta-shocks as limits of vanishing viscosity fo multidimensional zero-pressure gas dynamics, Quart. Appl. Math., 59 (2001), 315-342. [21] J. Li, Zhicheng Yang and Y. Zheng, Characteristic decompositions and interaction for rarefaction waves of the 2D Euler equations, to appear in J. Diff. Euqs., 2011. [22] J. Li and T. Zhang, Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations, in "Nonlinear PDE and Related Areas" (G. Q. Chen et al. Eds.), pp. 219-232, World Scientific, Singapore, 1998. [23] J. Li, T. Zhang and Y. 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. [24] J. Li and Y. Zheng, Interaction of rarefaction waves of the 2D self-similar Euler equations, Arch. Rat. Mech. Anal., 193 (2009), 623-657. doi: 10.1007/s00205-008-0140-6. [25] M. Li and Y. Zheng, Semi-hyperbolic patches of solutions of the 2D Euler equations, to appear in Arch. Rat. Mech. Anal., 2011. [26] T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems," John Wiley and Sons, 1994. [27] T. Li and W. Yu, "Boundary Value Problem for Quasilinear Hyperbolic Systems," Duke University, 1985. [28] D. Serre, Multi-dimensional shock interaction for a Chaplygin gas, Arch. Rat. Mech. Anal., 191 (2008), 539-577. doi: 10.1007/s00205-008-0110-z. [29] V. M. Shelkovich, $\delta$ and $\delta'$ wave types of singular solutions of systems of conservation laws and transport and concentration process, Russian Math. Surveys, 63 (2008), 473-546. doi: 10.1070/RM2008v063n03ABEH004534. [30] W. Sheng and T. Zhang, The Riemann problem for transportation equations in gas dynamics, Mem. Amer. Math. Soc. 137, 564 (1999). [31] K. Song and Y. Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system, Disc. Cont. Dyna. Syst., 24 (2009), 1365-1380. doi: 10.3934/dcds.2009.24.1365. [32] J. H. Spurk and N. Aksel, "Fluid Mechanics," Spring-Verlag Berlin Heidelberg, 2008. [33] D. Tan and T. Zhang and Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations, 112 (1994), 1-32. [34] T. Zhang and Y. Zheng, Conjecture on the structure of solution of the Riemann problem for 2D gas dynamics system, SIAM J. Math. Anal., 21 (1990), 593-630. doi: 10.1137/0521032. [35] Y. Zheng, "Systems of Conservation Laws: 2D Riemann Problems," 38 PNLDE, Birkhäuser, Boston, 2001. [36] Y. Zheng, Absorption of characteristics by sonic curves of the 2D Euler equations, Disc. Cont. Dyna. Syst., 23 (2009), 605-616. doi: 10.3934/dcds.2009.23.605.

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##### References:
 [1] S. Bang, Interaction of three and four rarefaction waves of the pressure-gradient system, J. Differential Equations, 246 (2009), 453-481. [2] Y. Brenier, Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations, J. Math. Fluid Mech., 7 (2005), 326-331. doi: 10.1007/s00021-005-0162-x. [3] S. X. Chen and A. F. Qu, Interaction of rarefaction waves in jet stream, J. Differential Equations, 248 (2010), 2931-2954. [4] G. Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanish pressure limit of solutions to the Euler equations for isentropic fliuds, SIAM J. Math. Anal., 34 (2003), 925-938. doi: 10.1137/S0036141001399350. [5] R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Interscience, New York, 1948. [6] Z. Dai ang T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, Arch. Rat. Mech. Anal., 155 (2000), 277-298. doi: 10.1007/s002050000113. [7] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, Springer-Verlag, 2003. [8] L. H. Guo, W. C. Sheng and T. Zhang, The 2D Riemann problem for isentropic Chaplygin gas dynamic system, Comm. Pure Appl. Anal., 9 (2010), 431-458. doi: 10.3934/cpaa.2010.9.431. [9] Y. B. Hu, J. Q. Li and W. C. Sheng, Interaction of rarefaction waves for 2D isothermal Euler equations, submitted for publication, 2011. [10] F. Huang and Z. Wang, Well posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146. doi: 10.1007/s002200100506. [11] F. John, "Partial Differential Equations," Springer-Verlag, 1982. [12] B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations, 118 (1995), 420-451. [13] D. J. Korchinski, "Solutions of a Riemann Problem for A 2 $\times$ 2 System of Conservation Laws Prosssing No Classical Solutions," thesis, Adelphi University, Garden City, NY, 1977. [14] N. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation, In "Nonlinear Functional Analysis and its Applications," Proc. Symp. Pure Math., 45 (1986), 81-90, Providence, Amer. Math. Soc.. [15] G. Lai and W. C. Sheng, Simple waves for 2D isentropic ir-rotational self-similar Euler system, Appl. Math. Mech, 31 (2010), 1-12. [16] P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. [17] Z. Lei and Y. Zheng, A complete global solution to the pressure gradient equation, J. Differential Equations, 236 (2007), 280-292. [18] J. Li, On the 2D gas expansion for compressible Euler euqations, SIAM J. Appl. Math., 62 (2001), 831-852. doi: 10.1137/S0036139900361349. [19] J. Li, Global solution of an initial-value problem for 2D compressible Euler equations, J. Differential Equations, 179 (2002), 178-194. [20] J. Li and H. Yang, Delta-shocks as limits of vanishing viscosity fo multidimensional zero-pressure gas dynamics, Quart. Appl. Math., 59 (2001), 315-342. [21] J. Li, Zhicheng Yang and Y. Zheng, Characteristic decompositions and interaction for rarefaction waves of the 2D Euler equations, to appear in J. Diff. Euqs., 2011. [22] J. Li and T. Zhang, Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations, in "Nonlinear PDE and Related Areas" (G. Q. Chen et al. Eds.), pp. 219-232, World Scientific, Singapore, 1998. [23] J. Li, T. Zhang and Y. 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. [24] J. Li and Y. Zheng, Interaction of rarefaction waves of the 2D self-similar Euler equations, Arch. Rat. Mech. Anal., 193 (2009), 623-657. doi: 10.1007/s00205-008-0140-6. [25] M. Li and Y. Zheng, Semi-hyperbolic patches of solutions of the 2D Euler equations, to appear in Arch. Rat. Mech. Anal., 2011. [26] T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems," John Wiley and Sons, 1994. [27] T. Li and W. Yu, "Boundary Value Problem for Quasilinear Hyperbolic Systems," Duke University, 1985. [28] D. Serre, Multi-dimensional shock interaction for a Chaplygin gas, Arch. Rat. Mech. Anal., 191 (2008), 539-577. doi: 10.1007/s00205-008-0110-z. [29] V. M. Shelkovich, $\delta$ and $\delta'$ wave types of singular solutions of systems of conservation laws and transport and concentration process, Russian Math. Surveys, 63 (2008), 473-546. doi: 10.1070/RM2008v063n03ABEH004534. [30] W. Sheng and T. Zhang, The Riemann problem for transportation equations in gas dynamics, Mem. Amer. Math. Soc. 137, 564 (1999). [31] K. Song and Y. Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system, Disc. Cont. Dyna. Syst., 24 (2009), 1365-1380. doi: 10.3934/dcds.2009.24.1365. [32] J. H. Spurk and N. Aksel, "Fluid Mechanics," Spring-Verlag Berlin Heidelberg, 2008. [33] D. Tan and T. Zhang and Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations, 112 (1994), 1-32. [34] T. Zhang and Y. Zheng, Conjecture on the structure of solution of the Riemann problem for 2D gas dynamics system, SIAM J. Math. Anal., 21 (1990), 593-630. doi: 10.1137/0521032. [35] Y. Zheng, "Systems of Conservation Laws: 2D Riemann Problems," 38 PNLDE, Birkhäuser, Boston, 2001. [36] Y. Zheng, Absorption of characteristics by sonic curves of the 2D Euler equations, Disc. Cont. Dyna. Syst., 23 (2009), 605-616. doi: 10.3934/dcds.2009.23.605.
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