Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems

doi:10.1007/s42967-022-00210-4

Abstract

Abstract: We are concerned with global solutions of multidimensional (M-D) Riemann problems for nonlinear hyperbolic systems of conservation laws, focusing on their global configurations and structures. We present some recent developments in the rigorous analysis of two-dimensional (2-D) Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations. In particular, we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.

Key words: Riemann problems, Two-dimensional (2-D), Transonic shocks, Solution structure, Free boundary problems, Mixed elliptic-hyperbolic type, Global configurations, Large-time asymptotics, Global attractors, Multidimensional (M-D), Shock capturing methods

CLC Number:

Gui-Qiang G. Chen. Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems[J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 1015-1052.

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References

[1] Agarwal, R., Halt, D.: A modified CUSP scheme in wave/particle split form for unstructured grid Euler flows. In: Caughey, D.A., Hafez, M.M. (eds.) Frontiers of Computational Fluid Dynamics, 155–163. World Scientific, Singapore (1994)
[2] Bae, M., Chen, G.-Q., Feldman, M.: Regularity of solutions to regular shock reflection for potential flow. Invent. Math. 175(3), 505–543 (2009)
[3] Bae, M., Chen, G.-Q., Feldman, M.: Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems, Research Monograph, 233 pages. Memoirs of the American Mathematical Society, Providence, RI (2023)
[4] Bargman, V.: On nearly glancing reflection of shocks. Office Sci. Res. and Develop. Rep. No. 5117 (1945)
[5] Ben-Dor, G.: Shock Wave Reflection Phenomena. Springer, New York (1991)
[6] Bressan, A., Chen, G.-Q., Lewicka, M., Wang, D.-H.: Nonlinear Conservation Laws and Applications. The IMA Volumes in Mathematics and Its Applications, vol. 153, Springer, New York (2011)
[7] 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(6), 1947–1977 (2006)
[8] Chang, T., Chen, G.-Q.: Diffraction of planar shock along the compressive corner. Acta Math. Sci. 6, 241–257 (1986)
[9] Chang, T., Chen, G.-Q., Yang, S.: 2-D Riemann problem in gas dynamics and formation of spiral. In: Nonlinear Problems in Engineering and Science—Numerical and Analytical Approach (Beijing, 1991), pp. 167–179, Science Press, Beijing (1992)
[10] Chang, T., Chen, G.-Q., Yang, S.: On the 2-D Riemann problem for the compressible Euler equations. I. Interaction of shocks and rarefaction waves. Discrete Contin. Dynam. Syst. 1, 555–584 (1995)
[11] Chang, T., Chen, G.-Q., Yang, S.: On the 2-D Riemann problem for the compressible Euler equations. II. Interaction of contact discontinuities. Discrete Contin. Dynam. Syst. 6, 419–430 (2000)
[12] Chang, T., Hsiao, L.: The Riemann problem and interaction of waves in gas dynamics. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc, New York (1989)
[13] Chen, G.-Q.: Euler equations and related hyperbolic conservation laws. Chapter 1. In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations, Evolutionary Equations, Vol. 2, Elsevier, Amsterdam (2005)
[14] Chen, G.-Q.: Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems. Sci. Chin. Math. 60(8), 1353–1370 (2017)
[15] Chen, G.-Q., Chen, J., Xiang, W.: Stability of attached transonic shocks in steady potential flow past three-dimensional wedges. Commun. Math. Phys. 387, 111–138 (2021)
[16] Chen, G.-Q., Cliffe, A., Huang, F., Liu, S., Wang, Q.: On the Riemann problem with four-shock interaction for the Euler equations for potential flow, Preprint (2022)
[17] Chen, G.-Q., Deng, X., Xiang, W.: Shock diffraction by convex cornered wedges for the nonlinear wave system. Arch. Ration. Mech. Anal. 211, 61–112 (2014)
[18] Chen, G.-Q., Fang, B.-X.: Stability of transonic shock-fronts in steady potential flow past a perturbed cone. Discrete Contin. Dyn. Syst. 23, 85–114 (2009)
[19] Chen, G.-Q., Fang, B.-X.: Stability of transonic shocks in steady supersonic flow past multidimensional wedges. Adv. Math. 314, 493–539 (2017)
[20] Chen, G.-Q., Feldman, M.: Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J. Am. Math. Soc. 16, 461–494 (2003)
[21] Chen, G.-Q., Feldman, M.: Global solutions to shock reflection by large-angle wedges for potential flow. Ann. Math. 171, 1019–1134 (2010)
[22] Chen, G.-Q., Feldman, M.: Mathematics of Shock Reflection-Diffraction and von Neumann’s Conjecture. Research Monograph, Annals of Mathematics Studies, 197. Princeton University Press, Princetion (2018)
[23] Chen, G.-Q., Feldman, M.: Multidimensional transonic shock waves and free boundary problems. Bull. Math. Sci. 12(1), Paper No. 2230002 (2022)
[24] Chen, G.-Q., Feldman, M., Hu, J., Xiang, W.: Loss of regularity of solutions of the shock diffraction problem by a convex cornered wedge for the potential flow equation. SIAM J. Math. 52(2), 1096–1114 (2020)
[25] Chen, G.-Q., Feldman, M., Xiang, W.: Convexity of self-similar transonic shock waves for potential flow. Arch. Ration. Mech. Anal. 238, 47–124 (2020)
[26] Chen, G.-Q., Feldman, M., Xiang, W.: Uniqueness of regular shock reflection/diffraction configurations for potential flow, Preprint (2022)
[27] Chen, G.-Q., Kuang, J., Zhang, Y.: Stability of conical shocks in the three-dimensional steady supersonic isothermal flows past Lipschitz perturbed cones. SIAM J. Math. Anal. 53, 2811–2862 (2021)
[28] Chen, G.-Q., LeFloch, P.: Entropy flux-splittings for hyperbolic conservation laws. Comm. Pure Appl. Math. 48, 691–729 (1995)
[29] Chen, G.-Q., Li, D., Tan, D.-C.: Structure of the Riemann solutions for two-dimensional scalar conservation laws. J. Differ. Equ. 127(1), 124–147 (1996)
[30] Chen, G.-Q., Shahgholian, H., Vázquez, J.-V.: Free boundary problems: the forefront of current and future developments. In: Free Boundary Problems and Related Topics. Theme Volume: Phil. Trans. R. Soc. A. 373, 20140285. The Royal Society, London (2015)
[31] Chen, G.-Q., Wang, Q., Zhu, S.-G.: Global solutions of a two-dimensional Riemann problem for the pressure gradient system. Comm. Pure Appl. Anal. 20, 2475–2503 (2021)
[32] Chen, S.-X.: Mathematical Analysis of Shock Wave Reflection. Series in Contemporary Mathematics 4, Shanghai Scientific and Technical Publishers, China; Springer Nature Singapore Pte Ltd., Singapore (2020)
[33] Chiodaroli, E., De Lellis, C., Kreml, O.: Global ill-posedness of the isentropic system of gas dynamics. Comm. Pure Appl. Math. 68, 1157–1190 (2015)
[34] Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Springer, New York (1948)
[35] Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. 4th edn. Springer, Berlin (2016)
[36] Elling, V.: Non-existence of strong regular reflections in self-similar potential flow. J. Differ. Eqs. 252, 2085–2103 (2012)
[37] Elling, V., Liu, T.-P.: Supersonic flow onto a solid wedge. Comm. Pure Appl. Math. 61, 1347–1448 (2008)
[38] Fletcher, C.H., Taub, A.H., Bleakney, W.: The Mach reflection of shock waves at nearly glancing incidence. Rev. Modern Phys. 23(3), 271–286 (1951)
[39] Fletcher, C.H., Weimer, D.K., Bleakney, W.: Pressure behind a shock wave diffracted through a small angle. Phys. Rev. 78(5), 634–635 (1950)
[40] Friedman, A.: Variational Principles and Free-Boundary Problems. 2nd edn. Robert E. Krieger Publishing Co., Inc., Malabar, Florida, (1988) [First edition, John Wiley & Sons, Inc., New York, 1982]
[41] Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. 2nd edn. Springer-Verlag, Berlin (1983)
[42] Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Anal. 18, 697–715 (1965)
[43] Glimm, J., Klingenberg, C., McBryan, O., Plohr, B., Sharp, D., Yaniv, S.: Front tracking and two-dimensional Riemann problems. Adv. Appl. Math. 6, 259–290 (1985)
[44] Glimm, J., Majda, A.: Multidimensional Hyperbolic Problems and Computations. The IMA Volumes in Mathematics and Its Applications, vol. 29. Springer, New York (1991)
[45] Guckenheimer, J.: Shocks and rarefactions in two space dimensions. Arch. Ration. Mech. Anal. 59, 281–291 (1975)
[46] Guderley, K.G.: The Theory of Transonic Flow. Translated from German by Moszynski, J.R. Pergamon Press, New York (1962)
[47] Harabetian, E.: Diffraction of a weak shock by a wedge. Comm. Pure Appl. Math. 40, 849–863 (1987)
[48] Hunter, J.K., Keller, J.B.: Weak shock diffraction. Wave Motion 6, 79–89 (1984)
[49] Keller, J.B., Blank, A.A.: Diffraction and reflection of pulses by wedges and corners. Comm. Pure Appl. Math. 4, 75–94 (1951)
[50] Kim, E.H.: A global sub-sonic solution to an interacting transonic shock of the self-similar nonlinear wave equation. J. Differ. Equ. 248, 2906–2930 (2010)
[51] Klingenberg, C., Kreml, O., Mácha, V., Markfelder, S.: Shocks make the Riemann problem for the full Euler system in multiple space dimensions ill-posed. Nonlinearity 33, 6517–6540 (2020)
[52] Kurganov, A., Tadmor, E.: Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer. Methods Partial Differ. Eqs. 18, 584–608 (2002)
[53] Lai, G., Sheng, W.: Two-dimensional pseudosteady flows around a sharp corner. Arch. Ration. Mech. Anal. 241, 805–884. (2021)
[54] Lax, P. D.: Shock waves and entropy. In: Zarantonllo, E.A. (ed.) Contributions to Nonlinear Functional Analysis, pp. 603–634. Academic Press, New York (1971)
[55] Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-RCSM. SIAM, Philiadelphia (1973)
[56] Lax, P.D., Liu, X.-D.: Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19, 319–340 (1998)
[57] LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser, Basel (1992)
[58] Li, J., Yang, Z., Zheng, Y.: Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations. J. Differ. Equ. 250, 782–798 (2011)
[59] Li, J., Zhang, T., Yang, S.: The Two-Dimensional Riemann Problem in Gas Dynamics. Longman Scientific & Technical, Harlow (1998)
[60] Li, J., Zheng, Y.: Interaction of rarefaction waves of the two-dimensional self-similar Euler equations. Arch. Ration. Mech. Anal. 193, 623–657 (2009)
[61] Li, J., Zheng, Y.: Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations. Commun. Math. Phys. 296, 303–321 (2010)
[62] Li, Y.F., Cao, Y.M.: Large-particle difference method with second-order accuracy in gas dynamics. Sci. Chin. 28A, 1024–1035 (1985)
[63] Lighthill, M.J.: The diffraction of a blast I. Proc. R. Soc. Lond. 198A, 454–470 (1949)
[64] Lighthill, M.J.: The diffraction of a blast II. Proc. R. Soc. Lond. 200A, 554–565 (1950)
[65] Lindquist, W.B.: Scalar Riemann problem in two spatial dimensions: piecewise smoothness of solutions and its breakdown. SIAM J. Math. Anal. 17, 1178–1197 (1986)
[66] Liu, T.-P.: Admissible Solutions of Hyperbolic Conservation Laws. Memoirs of the American Mathematical Society, 30(240), 1–78 (1981)
[67] Lock, G.D., Dewey, J.M.: An experimental investigation of the sonic criterion for transition from regular to Mach reflection of weak shock waves. Exp. Fluids 7, 289–292 (1989)
[68] Mach, E.: Über den verlauf von funkenwellenin der ebene und im raume. Sitzungsber. Akad. Wiss. Wien 78, 819–838 (1878)
[69] Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984)
[70] Menikoff, R., Plohr, B.: Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61, 75–130 (1989)
[71] Meyer, Th.: Über zweidimensionale Bewegungsvorgänge in einem Gas, das mit Überschallgeschwindigkeit strömt. Dissertation, Göttingen, 1908. Forschungsheft des Vereins deutscher Ingenieure, vol. 62, pp. 31–67, Berlin (1908)
[72] Morawetz, C.S.: Potential theory for regular and Mach reflection of a shock at a wedge. Comm. Pure Appl. Math. 47, 593–624 (1994)
[73] Prandtl, L.: Allgemeine Überlegungen über die Strömung zusammendrückbarer Fluüssigkeiten. Z. Angew. Math. Mech. 16, 129–142 (1938)
[74] Riemann, B.: Über die Fortpflanzung ebener Luftvellen von endlicher Schwingungsweite. Gött. Abh. Math. Cl. 8, 43–65 (1860)
[75] Schulz-Rinne, C.W.: Classification of the Riemann problem for two-dimensional gas dynamics. SIAM J. Math. Anal. 24, 76–88 (1993)
[76] Schulz-Rinne, C.W., Collins, J.P., Glaz, H.M.: Numerical solution of the Riemann problem for two-dimensional gas dynamics. SIAM J. Sci. Comput. 14, 1394–1414 (1993)
[77] Serre, D.: Shock reflection in gas dynamics. In: Handbook of Mathematical Fluid Dynamics, vol. 4, pp. 39–122. Elsevier, North-Holland (2007)
[78] Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Acta Numer. 29, 701–762 (2020)
[79] Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, New York (1982)
[80] Tan, D.C., Zhang, T.: Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws (I)–(II). J. Differ. Equ. 111, 203–282 (1994)
[81] Van Dyke, M.: An Album of Fluid Motion. The Parabolic Press, Stanford (1982)
[82] Von Neumann, J.: Theory of shock waves, Progress Report. U.S. Dept. Comm. Off. Tech. Serv. No. PB32719, Washington, DC (1943)
[83] Von Neumann, J.: Oblique reflection of shocks, Explo. Res. Rep. 12. Navy Department, Bureau of Ordnance, Washington, DC (1943)
[84] Von Neumann, J.: Refraction, intersection, and reflection of shock waves, NAVORD Rep. 203-45. Navy Department, Bureau of Ordnance, Washington, DC (1945)
[85] Von Neumann, J.: Collected Works, vol. 6. Pergamon, New York (1963)
[86] Von Neumann, J.: Discussion on the existence and uniqueness or multiplicity of solutions of the aerodynamical equation [Reprinted from MR0044302 (1949)]. Bull. Amer. Math. Soc. (N.S.), 47, 145–154 (2010)
[87] Wagner, D.H.: The Riemann problem in two space dimensions for a single conservation laws. SIAM J. Math. Anal. 14, 534–559 (1983)
[88] Wendroff, B.: The Riemann problem for materials with nonconvex equations of state: I. Isentropic flow. J. Math. Anal. Appl. 38, 454–466 (1972)
[89] Wendroff, B.: The Riemann problem for materials with nonconvex equations of state: II. General flow. J. Math. Anal. Appl. 38, 640–658 (1972)
[90] Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
[91] Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comp. Phys. 54, 115–173 (1984)
[92] Zhang, P., Li, J., Zhang, T.: On two-dimensional Riemann problem for pressure-gradient equations of the Euler system. Discrete Contin. Dynam. Systems 4, 609–634 (1998)
[93] Zhang, T., Zheng, Y.: Two-dimensional Riemann problem for a scalar conservation law. Trans. Am. Math. Soc. 312, 589–619 (1989)
[94] Zhang, T., Zheng, Y.: Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics. SIAM J. Math. Anal. 21, 593–630 (1990)
[95] Zheng, Y.: Existence of solutions to the transonic pressure gradient equations of the compressible Euler equations in elliptic regions. Commun. Partial Differ. Equ. 22, 1849–1868 (1997)
[96] Zheng, Y.: A global solution to a two-dimensional Riemann problem involving shocks as free boundaries. Acta Math. Appl. Sin. 19(4), 559–572 (2003)
[97] Zheng, Y.: Two-dimensional regular shock reflection for the pressure gradient system of conservation laws. Acta Math. Appl. Sin. 22(2), 177–210 (2006)
[98] Zheng, Y.: Systems of Conservation Laws: Two-Dimensional Riemann Problems, vol. 38. Springer, New York (2012)