Advertisement

Indian Journal of Pure and Applied Mathematics

, Volume 50, Issue 4, pp 1067–1086 | Cite as

Generalized plane delta shock waves for the n-dimensional zero-pressure gas dynamics with energy conservation law

  • Yanyan ZhangEmail author
  • Yu ZhangEmail author
Article
  • 14 Downloads

Abstract

By virtue of the generalized plane wave solution, we study a type of generalized plane delta shock wave for the n-dimensional zero-pressure gas dynamics governed by the conservation of mass, momentum and energy. It is found that a special kind of generalized plane delta shock wave on which both state variables simultaneously contain the Dirac delta functions appears in Riemann solutions, which is significantly different from the customary ones on which only one state variable contains the Dirac delta function. The generalized Rankine-Hugoniot relation of the generalized plane delta shock wave is derived. Under a suitable entropy condition, we further solve a kind of n-dimensional Riemann problem with Randon measure as initial data, and four different explicit configurations of solutions are constructively established. Finally, the overtaking of two plane delta shock waves is analyzed.

Key words

n-Dimensional zero-pressure gas dynamics energy conservation law generalized plane delta shock wave vacuum generalized Rankine-Hugoniot relation entropy condition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgement

Special gratefulness to the anonymous referee for his/her careful valuable suggestions, which have improved the original manuscript greatly.

References

  1. 1.
    R. K. Agarwal and D. W. Halt, A modified CUSP scheme in wave/particle split form for unstructured grid Euler flows. In: D.A. Caughey, M.M. Hafez(eds.) Frontiers of Computational Fluid Dynamics, 155–163. World Scientific, Singapore (1994).Google Scholar
  2. 2.
    S. Albeverio, O. S. Rozanova, and V. M. Shelkovich, Transport and concentration processes in the multidimensional zero-pressure gas dynamics model with the energy conservation law, arXiv:1101.5815v1, 2011.Google Scholar
  3. 3.
    F. Bouchut, On zero pressure gas dynamics, In: B. Perthame (ed.) Advances in Kinetic Theory and Computing, Series on Advances in Mathematics for Applied Sciences, 22 (1994), 171–190. World Scientific, Singapore.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317–2328.MathSciNetCrossRefGoogle Scholar
  5. 5.
    W. Cai and Y. Zhang, Interactions of delta shock waves for zero-pressure gas dynamics with energy conservation law, Advances in Mathematical Physics, 2016, Article ID 1783689, 12 pages.Google Scholar
  6. 6.
    G. Chen and H. Liu, Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations, SIAM J. Math. Anal., 34 (2003), 925–938.MathSciNetCrossRefGoogle Scholar
  7. 7.
    H. Cheng, Riemann problem of one-dimensional system of conservation laws of mass, momentum and energy in zero-pressure gas dynamics, Differ. Equ. Appl., 4(4) (2012), 653–664.MathSciNetzbMATHGoogle Scholar
  8. 8.
    R. Courant and D. Hilbert, Methods of mathematical physics, Interscience, New York (1962).zbMATHGoogle Scholar
  9. 9.
    L. Guo, L. Pan, and G. Yin, The perturbed Riemann problem and delta contact discontinuity in chromatography equations, Nonlinear Analysis: Theory, Methods & Applications, 106 (2014), 110–123.MathSciNetCrossRefGoogle Scholar
  10. 10.
    L. Guo and G. Yin, The Riemann problem with delta initial data for the one-dimensional transport equations, B. Malays. Math. Sci. So., 38(1) (2015), 219–230.MathSciNetCrossRefGoogle Scholar
  11. 11.
    L. Guo, Y. Zhang, and G. Yin, Interactions of delta shock waves for the Chaplygin gas equations with split delta functions, J. Math. Anal. Appl., 410(1) (2014), 190–201.MathSciNetCrossRefGoogle Scholar
  12. 12.
    F. Huang and Z. Wang, Well-posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117–146.MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. Kraiko, Discontinuity surfaces in medium without self-pressure, Priklad. Mat. I Mekhan., 43 (1979), 539–549.Google Scholar
  14. 14.
    P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, Philadelphia, 1973.Google Scholar
  15. 15.
    J. Li and H. Yang, Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure gas dynamics, Q. Appl. Math., 59(2) (2001), 315–342.MathSciNetCrossRefGoogle Scholar
  16. 16.
    J. Li, S. Yang, and T. Zhang, The two-dimensional Riemann problem in gas dynamics, Longman, London (1998).zbMATHGoogle Scholar
  17. 17.
    J. Li and T. Zhang, Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations, Advances in nonlinear partial differential equations and related areas, World Sci. Publishing, River Edge, NJ, 1998.CrossRefGoogle Scholar
  18. 18.
    Y. Li and Y. Cao, Second order large particle difference method, Sci. China Ser. A, 8 (1985), 1024–1035 (in Chinese).zbMATHGoogle Scholar
  19. 19.
    M. Nedeljkov, L. Neumann, M. Oberguggenberger, and M. R. Sahoo, Radially symmetric shadow wave solutions to the system of pressureless gas dynamics in arbitrary dimensions, Nonlinear Analysis: Theory, Methods & Applications, 163 (2017), 104–126.MathSciNetCrossRefGoogle Scholar
  20. 20.
    B. Nilsson, O. Rozanova, and V. M. Shelkovich, Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks: II, Applicable Analysis, 90(5) (2011), 831–842.MathSciNetCrossRefGoogle Scholar
  21. 21.
    B. Nilsson and V. M. Shelkovich, Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks, Applicable Analysis, 90(11) (2011), 1677–1689.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Y. Pang, Delta shock wave in the compressible Euler equations for a Chaplygin gas, J. Math. Anal. Appl., 448(1) (2017), 245–261.MathSciNetCrossRefGoogle Scholar
  23. 23.
    S. F. Shandarin and Ya. B. Zeldovich, The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium, Rev. Modern Phys., 61 (1989), 185–220.MathSciNetCrossRefGoogle Scholar
  24. 24.
    V. M. Shelkovich, The Riemann problem admitting δ, δ′-shocks, and vacuum states (the vanishing viscosity approach), J. Differential Equations, 231 (2006), 459–500.MathSciNetCrossRefGoogle Scholar
  25. 25.
    C. Shen and M. Sun, Interactions of delta shock waves for the transport equations with split delta functions, J. Math. Anal. Appl., 351(2) (2009), 747–755.MathSciNetCrossRefGoogle Scholar
  26. 26.
    C. Shen and M. Sun, Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws, Nonlinear Analysis: Theory, Methods & Applications, 73(10) (2010), 3284–3294.MathSciNetCrossRefGoogle Scholar
  27. 27.
    W. Sheng and T. Zhang, The Riemann problem for transportation equation in gas dynamics, Mem. Am. Math. Soc., 137(654) (1999), 1–77.MathSciNetGoogle Scholar
  28. 28.
    D. Tan and T. Zhang, Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws I. Four-J cases, II. Initial data involving some rarefaction waves, J. Differential Equations, 111 (1994), 203–282.MathSciNetCrossRefGoogle Scholar
  29. 29.
    D. Tan, 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.MathSciNetCrossRefGoogle Scholar
  30. 30.
    L. Wang, The Riemann problem with delta data for zero-pressure gas dynamics, Chinese Annals of Mathematics, Series B, 37(3) (2016), 441–450.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Z. Wang and Q. Zhang, The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations, Acta Math. Sci., 32(3) (2012), 825–841.MathSciNetCrossRefGoogle Scholar
  32. 32.
    E. Weinan, Yu. G. Rykov, and Ya. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in ashesion particle dynamics, Comm. Math. Phys., 177 (1996), 349–380.MathSciNetCrossRefGoogle Scholar
  33. 33.
    H. Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differential Equations, 159 (1999), 447–484.MathSciNetCrossRefGoogle Scholar
  34. 34.
    H. Yang, Generalized plane delta-shock waves for n-dimensional zero-pressure gas dynamics, J. Math. Anal. Appl., 260 (2001), 18–35.MathSciNetCrossRefGoogle Scholar
  35. 35.
    H. Yang and W. Sun, The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws, Nonlinear Analysis: Theory, Methods & Applications, 67(11) (2007), 3041–3049.MathSciNetCrossRefGoogle Scholar
  36. 36.
    H. Yang and Y. Zhang, New developments of delta shock waves and its applications in systems of conservation laws, J. Differential Equations, 252 (2012), 5951–5993.MathSciNetCrossRefGoogle Scholar
  37. 37.
    H. Yang and Y. Zhang, Delta shock waves with Dirac delta function in both components for systems of conservation laws, J. Differential Equations, 257 (2014), 4369–4402.MathSciNetCrossRefGoogle Scholar
  38. 38.
    Y. Zhang and Y. Zhang, Vanishing viscosity limit for Riemann solutions to a class of non-strictly hyperbolic systems, Acta Appl. Math., 155 (2018), 151–175.MathSciNetCrossRefGoogle Scholar
  39. 39.
    Y. Zhang and Y. Zhang, The Riemann problem and interaction of waves in two-dimensional steady zero-pressure adiabatic flow, International Journal of Non-Linear Mechanics, 104 (2018), 100–108.CrossRefGoogle Scholar
  40. 40.
    Y. Zhang and Y. Zhang, Viscous limits for a Riemannian problem to a class of systems of conservation laws, Rocky Mountain Journal of Mathematics, 48(5) (2018), 1721–1741.MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian National Science Academy 2019

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsXinyang Normal UniversityXinyangPeople’s Republic of China
  2. 2.Department of MathematicsYunnan Normal UniversityKunmingPeople’s Republic of China

Personalised recommendations