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 & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

Y. Brenier, Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations,, J. Math. Fluid Mech., 7 (2005), 326. doi: 10.1007/s00021-005-0162-x. Google Scholar

[3]

S. X. Chen and A. F. Qu, Interaction of rarefaction waves in jet stream,, J. Differential Equations, 248 (2010), 2931. Google Scholar

[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. doi: 10.1137/S0036141001399350. Google Scholar

[5]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves,", Interscience, (1948). Google Scholar

[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. doi: 10.1007/s002050000113. Google Scholar

[7]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics in Mathematics, (2003). Google Scholar

[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. doi: 10.3934/cpaa.2010.9.431. Google Scholar

[9]

Y. B. Hu, J. Q. Li and W. C. Sheng, Interaction of rarefaction waves for 2D isothermal Euler equations,, submitted for publication, (2011). Google Scholar

[10]

F. Huang and Z. Wang, Well posedness for pressureless flow,, Comm. Math. Phys., 222 (2001), 117. doi: 10.1007/s002200100506. Google Scholar

[11]

F. John, "Partial Differential Equations,", Springer-Verlag, (1982). Google Scholar

[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. Google Scholar

[13]

D. J. Korchinski, "Solutions of a Riemann Problem for A 2 $\times$ 2 System of Conservation Laws Prosssing No Classical Solutions,", thesis, (1977). Google Scholar

[14]

N. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation,, In, 45 (1986), 81. Google Scholar

[15]

G. Lai and W. C. Sheng, Simple waves for 2D isentropic ir-rotational self-similar Euler system,, Appl. Math. Mech, 31 (2010), 1. Google Scholar

[16]

P. Lax, Hyperbolic systems of conservation laws II,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. Google Scholar

[17]

Z. Lei and Y. Zheng, A complete global solution to the pressure gradient equation,, J. Differential Equations, 236 (2007), 280. Google Scholar

[18]

J. Li, On the 2D gas expansion for compressible Euler euqations,, SIAM J. Appl. Math., 62 (2001), 831. doi: 10.1137/S0036139900361349. Google Scholar

[19]

J. Li, Global solution of an initial-value problem for 2D compressible Euler equations,, J. Differential Equations, 179 (2002), 178. Google Scholar

[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. Google Scholar

[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). Google Scholar

[22]

J. Li and T. Zhang, Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations,, in, (1998), 219. Google Scholar

[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. doi: 10.1007/s00220-006-0033-1. Google Scholar

[24]

J. Li and Y. Zheng, Interaction of rarefaction waves of the 2D self-similar Euler equations,, Arch. Rat. Mech. Anal., 193 (2009), 623. doi: 10.1007/s00205-008-0140-6. Google Scholar

[25]

M. Li and Y. Zheng, Semi-hyperbolic patches of solutions of the 2D Euler equations,, to appear in Arch. Rat. Mech. Anal., (2011). Google Scholar

[26]

T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems,", John Wiley and Sons, (1994). Google Scholar

[27]

T. Li and W. Yu, "Boundary Value Problem for Quasilinear Hyperbolic Systems,", Duke University, (1985). Google Scholar

[28]

D. Serre, Multi-dimensional shock interaction for a Chaplygin gas,, Arch. Rat. Mech. Anal., 191 (2008), 539. doi: 10.1007/s00205-008-0110-z. Google Scholar

[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. doi: 10.1070/RM2008v063n03ABEH004534. Google Scholar

[30]

W. Sheng and T. Zhang, The Riemann problem for transportation equations in gas dynamics,, Mem. Amer. Math. Soc. 137, 564 (1999). Google Scholar

[31]

K. Song and Y. Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system,, Disc. Cont. Dyna. Syst., 24 (2009), 1365. doi: 10.3934/dcds.2009.24.1365. Google Scholar

[32]

J. H. Spurk and N. Aksel, "Fluid Mechanics,", Spring-Verlag Berlin Heidelberg, (2008). Google Scholar

[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. Google Scholar

[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. doi: 10.1137/0521032. Google Scholar

[35]

Y. Zheng, "Systems of Conservation Laws: 2D Riemann Problems,", 38 PNLDE, (2001). Google Scholar

[36]

Y. Zheng, Absorption of characteristics by sonic curves of the 2D Euler equations,, Disc. Cont. Dyna. Syst., 23 (2009), 605. doi: 10.3934/dcds.2009.23.605. Google Scholar

show all references

References:
[1]

S. Bang, Interaction of three and four rarefaction waves of the pressure-gradient system,, J. Differential Equations, 246 (2009), 453. Google Scholar

[2]

Y. Brenier, Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations,, J. Math. Fluid Mech., 7 (2005), 326. doi: 10.1007/s00021-005-0162-x. Google Scholar

[3]

S. X. Chen and A. F. Qu, Interaction of rarefaction waves in jet stream,, J. Differential Equations, 248 (2010), 2931. Google Scholar

[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. doi: 10.1137/S0036141001399350. Google Scholar

[5]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves,", Interscience, (1948). Google Scholar

[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. doi: 10.1007/s002050000113. Google Scholar

[7]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics in Mathematics, (2003). Google Scholar

[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. doi: 10.3934/cpaa.2010.9.431. Google Scholar

[9]

Y. B. Hu, J. Q. Li and W. C. Sheng, Interaction of rarefaction waves for 2D isothermal Euler equations,, submitted for publication, (2011). Google Scholar

[10]

F. Huang and Z. Wang, Well posedness for pressureless flow,, Comm. Math. Phys., 222 (2001), 117. doi: 10.1007/s002200100506. Google Scholar

[11]

F. John, "Partial Differential Equations,", Springer-Verlag, (1982). Google Scholar

[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. Google Scholar

[13]

D. J. Korchinski, "Solutions of a Riemann Problem for A 2 $\times$ 2 System of Conservation Laws Prosssing No Classical Solutions,", thesis, (1977). Google Scholar

[14]

N. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation,, In, 45 (1986), 81. Google Scholar

[15]

G. Lai and W. C. Sheng, Simple waves for 2D isentropic ir-rotational self-similar Euler system,, Appl. Math. Mech, 31 (2010), 1. Google Scholar

[16]

P. Lax, Hyperbolic systems of conservation laws II,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. Google Scholar

[17]

Z. Lei and Y. Zheng, A complete global solution to the pressure gradient equation,, J. Differential Equations, 236 (2007), 280. Google Scholar

[18]

J. Li, On the 2D gas expansion for compressible Euler euqations,, SIAM J. Appl. Math., 62 (2001), 831. doi: 10.1137/S0036139900361349. Google Scholar

[19]

J. Li, Global solution of an initial-value problem for 2D compressible Euler equations,, J. Differential Equations, 179 (2002), 178. Google Scholar

[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. Google Scholar

[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). Google Scholar

[22]

J. Li and T. Zhang, Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations,, in, (1998), 219. Google Scholar

[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. doi: 10.1007/s00220-006-0033-1. Google Scholar

[24]

J. Li and Y. Zheng, Interaction of rarefaction waves of the 2D self-similar Euler equations,, Arch. Rat. Mech. Anal., 193 (2009), 623. doi: 10.1007/s00205-008-0140-6. Google Scholar

[25]

M. Li and Y. Zheng, Semi-hyperbolic patches of solutions of the 2D Euler equations,, to appear in Arch. Rat. Mech. Anal., (2011). Google Scholar

[26]

T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems,", John Wiley and Sons, (1994). Google Scholar

[27]

T. Li and W. Yu, "Boundary Value Problem for Quasilinear Hyperbolic Systems,", Duke University, (1985). Google Scholar

[28]

D. Serre, Multi-dimensional shock interaction for a Chaplygin gas,, Arch. Rat. Mech. Anal., 191 (2008), 539. doi: 10.1007/s00205-008-0110-z. Google Scholar

[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. doi: 10.1070/RM2008v063n03ABEH004534. Google Scholar

[30]

W. Sheng and T. Zhang, The Riemann problem for transportation equations in gas dynamics,, Mem. Amer. Math. Soc. 137, 564 (1999). Google Scholar

[31]

K. Song and Y. Zheng, Semi-hyperbolic patches of solutions of the pressure gradient system,, Disc. Cont. Dyna. Syst., 24 (2009), 1365. doi: 10.3934/dcds.2009.24.1365. Google Scholar

[32]

J. H. Spurk and N. Aksel, "Fluid Mechanics,", Spring-Verlag Berlin Heidelberg, (2008). Google Scholar

[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. Google Scholar

[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. doi: 10.1137/0521032. Google Scholar

[35]

Y. Zheng, "Systems of Conservation Laws: 2D Riemann Problems,", 38 PNLDE, (2001). Google Scholar

[36]

Y. Zheng, Absorption of characteristics by sonic curves of the 2D Euler equations,, Disc. Cont. Dyna. Syst., 23 (2009), 605. doi: 10.3934/dcds.2009.23.605. Google Scholar

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