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Interactions of Rarefaction Waves and Rarefaction Shocks of the Two-Dimensional Compressible Euler Equations with General Equation of State

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Abstract

We consider two types of wave interactions arising from the expansion of a wedge of gas to vacuum: interaction of rarefaction waves and interaction of shock-rarefaction composite waves. In order to construct the solutions to these wave interactions, we consider a standard Goursat problem and a discontinuous Goursat problem for the two-dimensional self-similar Euler system. Global solutions to these Goursat problems are constructed by the method of characteristics. The main difficulty to construct the global solutions is that the type of the 2D self-similar Euler system is a priori unknown. By constructing invariant regions of the characteristic angles, we obtain the hyperbolicity of the system in the interaction regions. The solutions constructed in this paper can be used as building blocks of solutions to 2D Riemann problems of the Euler equations with general equations of state.

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References

  1. Bae, M., Chen, G.Q., Feldman, M.: Prandtl-Meyer reflection for supersonic flow past a solid ramp. Quart. Appl. Math. 71, 583–600 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Canic, S., Keyfitz, B.L.: Quasi-one-dimensional Riemann problems and their role in self-similar two-dimensional problems. Arch. Ration. Mech. Anal. 144, 233–258 (1998)

    MathSciNet  MATH  Google Scholar 

  3. Canic, S., Keyfitz, B.L., Kim, E.H.: A free boundary problem for a quasilinear degenerate elliptic equation: regular reflection of weak shock. Commun. Pure Appl. Math. 55, 71–92 (2002)

    MATH  Google Scholar 

  4. Canic, S., Keyfitz, B.L., Kim, E.H.: Free boundary problems for nonlinear wave systems: Mach stems for interacting shocks. SIAM J. Math. Anal. 37, 1947–1977 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Chang, T., Chen, G.Q., Yang, S.L.: On the 2-D Riemann problem for the compressible Euler equations, I. Interaction of shock waves and rarefaction waves. Discret. Contin. Dyn. Syst. 1, 555–584 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Chang, T., Chen, G.Q., Yang, S.L.: On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities. Discret. Contin. Dyn. Syst. 6, 419–430 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Chen, G.Q., Feldman, M.: Global solutions of shock reflection by large-angle wedges for potential flow. Ann. Math. 171, 1067–1182 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Chen, G.Q., Feldman, M.: The Mathematics of Shock Reflection-Diffraction and von Neumanns Conjectures. Mathematics Series in Annals of Mathematical Studies, vol. 197. Princeton University Press, Princeton (2017)

    Google Scholar 

  9. Chen, G.Q., Feldman, M., Xiang, W.: Convexity of self-similar transonic shocks and free boundaries for the Euler equations for potential flow. Arch. Ration. Mech. Anal. 238, 47–124 (2020)

    MathSciNet  MATH  Google Scholar 

  10. Chen, G.Q., Deng, X.M., Xiang, W.: Shock diffraction by convex cornered wedges for the nonlinear wave system. Arch. Ration. Mech. Anal. 211, 61–112 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Chen, S.X., Qu, A.F.: Riemann boundary value problems and reflection of shock for the Chaplygin gas. Sci. China Math. 55, 671–685 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Chen, S.X., Qu, A.F.: Two dimensional Riemann problems for Chaplygin gas. SIAM J. Math. Anal. 44, 2146–2178 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Chen, X., Zheng, Y.X.: The interaction of rarefaction waves of the two-dimensional Euler equations. Indiana Univ. Math. J. 59, 231–256 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience, New York (1948)

    MATH  Google Scholar 

  15. Dai, Z.H., Zhang, T.: Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics. Arch. Ration. Mech. Anal. 155, 277–298 (2000)

    MathSciNet  MATH  Google Scholar 

  16. Elling, V.: Regular reflection in self-similar potential flow and the sonic criterion. Commun. Math. Anal. 8, 22–69 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Elling, V., Liu, T.P.: Supersonic flow onto a solid wedge. Commun. Pure Appl. Math. 61, 1347–1448 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Fossati, M., Quartapelle, L.: The Reimann problem for hyperbolic equations under a nonconvex flux with two inflection points. arXiv: 1402.5906

  19. Glimm, J., Ji, X.M., Li, J.Q., Li, X.L., Zhang, P., Zhang, T., Zheng, Y.X.: Transonic shock formation in a rarefaction Riemann problem for the 2D compressible Euler equations. SIAM J. Appl. Math. 69, 720–742 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Godlewski, E., Seguin, N.: The Riemann problem for a simple model of phase transition. Commun. Math. Sci. 4, 227–247 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Ji, X.M., Zheng, Y.X.: Characteristic decouplings and interactions of rarefaction waves of 2D Euler equations. J. Math. Anal. Appl. 406, 4–14 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Lai, G.: On the expansion of a wedge of van der Waals gas into a vacuum. J. Differ. Equ. 259, 1181–1202 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Lai, G.: On the expansion of a wedge of van der Waals gas into a vacuum II. J. Differ. Equ. 260, 3538–3575 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Lai, G.: Global solutions to a class of two-dimensional Riemann problems for the isentropic Euler equations with general equations of state. Indian. Univ. Math. J. 68, 1409–1464 (2019)

    MATH  Google Scholar 

  25. LeFloch, P.G.: Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. Lectures in Mathematics-ETH Zürich. Birkhäuser Verlag, Basel (2002)

    Google Scholar 

  26. Li, J.Q.: On the two-dimensional gas expansion for the compressible Euler equations. SIAM J. Appl. Math. 62, 831–852 (2002)

    MathSciNet  MATH  Google Scholar 

  27. Li, J.Q., Yang, Z.C., Zheng, Y.X.: Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations. J. Differ. Equ. 250, 782–798 (2011)

    MathSciNet  MATH  Google Scholar 

  28. Li, J.Q., Zhang, T., Yang, S.L.: The Two-Dimensional Riemann Problem in Gas Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 98. Longman (1998)

  29. Li, J.Q., Zhang, T., Zheng, Y.X.: Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations. Commun. Math. Phys. 267, 1–12 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Li, J.Q., Zheng, Y.X.: Interaction of rarefaction waves of the two-dimensional self-similar Euler equations. Arch. Ration. Mech. Anal. 193, 623–657 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Li, J.Q., Zheng, Y.X.: Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations. Commun. Math. Phys. 296, 303–321 (2010)

    MathSciNet  MATH  Google Scholar 

  32. Li, M.J., Zheng, Y.X.: Semi-hyperbolic patches of solutions of the two-dimensional Euler equations. Arch. Ration. Mech. Anal. 201, 1069–1096 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Li, T.T., Yu, W.C.: Boundary Value Problem for Quasilinear Hyperbolic Systems. Duke University (1985)

  34. Li, T.T., Yu, W.C.: The centered waves for quasilinear hyperbolic systems. Fudan (J. Nat. Sci.) 25, 195–206 (1986). in Chinese

    MathSciNet  MATH  Google Scholar 

  35. Li, T.T., Yu, W.C.: The centered wave problem for quasilinear hyperbolic systems. Chin. Ann. Math. 7A, 423–436 (1986). (in Chinese)

    MATH  Google Scholar 

  36. Menikoff, R., Plohr, B.J.: The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61, 75–130 (1989)

    MathSciNet  MATH  Google Scholar 

  37. Serre, D.: Shock reflection in gas dynamics. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathemathematical Fluid Dynamics, vol. IV, pp. 39–122. North-Holland, Elsevier (2007)

    Google Scholar 

  38. Serre, D.: Multi-dimensional shock interaction for a Chaplygin gas. Arch. Ration. Mech. Anal. 191, 539–577 (2008)

    MATH  Google Scholar 

  39. Serre, D.: Expansion of a compressible gas in vacuum. Bull. Inst. Math. Acad. Sin. (N.S.) 10, 695–716 (2015)

    MathSciNet  MATH  Google Scholar 

  40. Sheng, W.C., Zhang, T.: The Riemann problem for transportation equations in gas dynamics. Mem. Am. Math. Soc. 137, 564 (1999)

    MathSciNet  Google Scholar 

  41. Smoller, J.: Shock Waves and Reaction Diffusion Equations, 2nd edn. Springer, New York (1994)

    MATH  Google Scholar 

  42. Suchkow, V.A.: Flow into a vacuum along an oblique wall. J. Appl. Math. Mech. 27, 1132–1134 (1963)

    Google Scholar 

  43. Wendroff, B.: The Riemann problem for materials with nonconvex equation of state, I. Isentropic flow. J. Math. Anal. Appl. 38, 454–466 (1972)

    MathSciNet  MATH  Google Scholar 

  44. Ying, L.A., Wang, C.H.: The discontinuous initial value problem of a reacting gas flow system. Trans. Am. Math. Soc. 266, 361–387 (1981)

    MATH  Google Scholar 

  45. Zafar, M., Sharma, V.D.: Expansion of a wedge of non-ideal gas into vacuum. Nonlinear Anal. Real World Appl. 31, 580–592 (2016)

    MathSciNet  MATH  Google Scholar 

  46. Zhang, T., Zheng, Y.X.: Conjecture on the structure of solution of the Riemann problem for 2D gas dynamics system. SIAM J. Math. Anal. 21, 593–630 (1990)

    MathSciNet  MATH  Google Scholar 

  47. Zhao, W.X.: The expansion of gas from a wedge with small angle into a vacuum. Commun. Pure Appl. Anal. 12, 2319–2330 (2013)

    MathSciNet  MATH  Google Scholar 

  48. Zheng, Y.X.: Systems of Conservation Laws: 2-D Riemann Problems. Progress in Nonlinear Differential Equations and Their Applications 38. \(Bikh\ddot{a}user\), Boston (2001)

  49. Zheng, Y.: Two-dimensional regular shock reflection for the pressure gradient system of conservation Laws. Acta Math. Appl. Sin. (Engl. Ser.) 22, 177–210 (2006)

    MathSciNet  MATH  Google Scholar 

  50. Zheng, Y.X.: Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discret. Contin. Dyn. Syst. 23, 605–616 (2009)

    MathSciNet  MATH  Google Scholar 

  51. Zhou, X.: The centered wave solution with large amplitude for quasilinear hyperbolic systems. Northeast. Math. J. 3, 439–451 (1987). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  52. Zhou, X.: The local solution with large amplitude of the centered wave problem for quasilinear hyperbolic systems. Chin. Ann. Math. 10A, 119–138 (1989). (in Chinese)

    MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Geng Lai

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Lai, G. Interactions of Rarefaction Waves and Rarefaction Shocks of the Two-Dimensional Compressible Euler Equations with General Equation of State. J Dyn Diff Equat 35, 381–419 (2023). https://doi.org/10.1007/s10884-021-09999-9

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