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

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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.

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References

  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. 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. 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.

    MathSciNet  MATH  Google Scholar 

  4. Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317–2328.

    Article  MathSciNet  Google Scholar 

  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. 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.

    Article  MathSciNet  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  8. R. Courant and D. Hilbert, Methods of mathematical physics, Interscience, New York (1962).

    MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  12. F. Huang and Z. Wang, Well-posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117–146.

    Article  MathSciNet  Google Scholar 

  13. A. Kraiko, Discontinuity surfaces in medium without self-pressure, Priklad. Mat. I Mekhan., 43 (1979), 539–549.

    Google Scholar 

  14. P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, Philadelphia, 1973.

    Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  16. J. Li, S. Yang, and T. Zhang, The two-dimensional Riemann problem in gas dynamics, Longman, London (1998).

    MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  18. Y. Li and Y. Cao, Second order large particle difference method, Sci. China Ser. A, 8 (1985), 1024–1035 (in Chinese).

    MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  22. Y. Pang, Delta shock wave in the compressible Euler equations for a Chaplygin gas, J. Math. Anal. Appl., 448(1) (2017), 245–261.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  27. W. Sheng and T. Zhang, The Riemann problem for transportation equation in gas dynamics, Mem. Am. Math. Soc., 137(654) (1999), 1–77.

    MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  33. H. Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differential Equations, 159 (1999), 447–484.

    Article  MathSciNet  Google Scholar 

  34. H. Yang, Generalized plane delta-shock waves for n-dimensional zero-pressure gas dynamics, J. Math. Anal. Appl., 260 (2001), 18–35.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

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Acknowledgement

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

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Authors and Affiliations

  1. College of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, People’s Republic of China

    Yanyan Zhang

  2. Department of Mathematics, Yunnan Normal University, Kunming, 650091, People’s Republic of China

    Yu Zhang

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  1. Yanyan Zhang
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  2. Yu Zhang
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Correspondence to Yanyan Zhang or Yu Zhang.

Additional information

Supported by National Natural Science Foundation of China (11501488), the Scientific Research Foundation of Xinyang Normal University (No. 0201318), Nan Hu Young Scholar Supporting Program of XYNU, the Scientific Research Foundation Project of Yunnan Education Department (2018JS150) and Yunnan Applied Basic Research Projects (2018FD015).

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Zhang, Y., Zhang, Y. Generalized plane delta shock waves for the n-dimensional zero-pressure gas dynamics with energy conservation law. Indian J Pure Appl Math 50, 1067–1086 (2019). https://doi.org/10.1007/s13226-019-0374-z

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  • DOI: https://doi.org/10.1007/s13226-019-0374-z

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