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.
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References
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)
Bae, M., Chen, G.-Q., Feldman, M.: Regularity of solutions to regular shock reflection for potential flow. Invent. Math. 175(3), 505–543 (2009)
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)
Bargman, V.: On nearly glancing reflection of shocks. Office Sci. Res. and Develop. Rep. No. 5117 (1945)
Ben-Dor, G.: Shock Wave Reflection Phenomena. Springer, New York (1991)
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)
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)
Chang, T., Chen, G.-Q.: Diffraction of planar shock along the compressive corner. Acta Math. Sci. 6, 241–257 (1986)
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)
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)
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)
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)
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)
Chen, G.-Q.: Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems. Sci. Chin. Math. 60(8), 1353–1370 (2017)
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)
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)
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)
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)
Chen, G.-Q., Fang, B.-X.: Stability of transonic shocks in steady supersonic flow past multidimensional wedges. Adv. Math. 314, 493–539 (2017)
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)
Chen, G.-Q., Feldman, M.: Global solutions to shock reflection by large-angle wedges for potential flow. Ann. Math. 171, 1019–1134 (2010)
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)
Chen, G.-Q., Feldman, M.: Multidimensional transonic shock waves and free boundary problems. Bull. Math. Sci. 12(1), Paper No. 2230002 (2022)
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)
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)
Chen, G.-Q., Feldman, M., Xiang, W.: Uniqueness of regular shock reflection/diffraction configurations for potential flow, Preprint (2022)
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)
Chen, G.-Q., LeFloch, P.: Entropy flux-splittings for hyperbolic conservation laws. Comm. Pure Appl. Math. 48, 691–729 (1995)
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)
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)
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)
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)
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)
Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Springer, New York (1948)
Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. 4th edn. Springer, Berlin (2016)
Elling, V.: Non-existence of strong regular reflections in self-similar potential flow. J. Differ. Eqs. 252, 2085–2103 (2012)
Elling, V., Liu, T.-P.: Supersonic flow onto a solid wedge. Comm. Pure Appl. Math. 61, 1347–1448 (2008)
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)
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)
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]
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. 2nd edn. Springer-Verlag, Berlin (1983)
Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Anal. 18, 697–715 (1965)
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)
Glimm, J., Majda, A.: Multidimensional Hyperbolic Problems and Computations. The IMA Volumes in Mathematics and Its Applications, vol. 29. Springer, New York (1991)
Guckenheimer, J.: Shocks and rarefactions in two space dimensions. Arch. Ration. Mech. Anal. 59, 281–291 (1975)
Guderley, K.G.: The Theory of Transonic Flow. Translated from German by Moszynski, J.R. Pergamon Press, New York (1962)
Harabetian, E.: Diffraction of a weak shock by a wedge. Comm. Pure Appl. Math. 40, 849–863 (1987)
Hunter, J.K., Keller, J.B.: Weak shock diffraction. Wave Motion 6, 79–89 (1984)
Keller, J.B., Blank, A.A.: Diffraction and reflection of pulses by wedges and corners. Comm. Pure Appl. Math. 4, 75–94 (1951)
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)
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)
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)
Lai, G., Sheng, W.: Two-dimensional pseudosteady flows around a sharp corner. Arch. Ration. Mech. Anal. 241, 805–884. (2021)
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)
Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-RCSM. SIAM, Philiadelphia (1973)
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)
LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser, Basel (1992)
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)
Li, J., Zhang, T., Yang, S.: The Two-Dimensional Riemann Problem in Gas Dynamics. Longman Scientific & Technical, Harlow (1998)
Li, J., Zheng, Y.: Interaction of rarefaction waves of the two-dimensional self-similar Euler equations. Arch. Ration. Mech. Anal. 193, 623–657 (2009)
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)
Li, Y.F., Cao, Y.M.: Large-particle difference method with second-order accuracy in gas dynamics. Sci. Chin. 28A, 1024–1035 (1985)
Lighthill, M.J.: The diffraction of a blast I. Proc. R. Soc. Lond. 198A, 454–470 (1949)
Lighthill, M.J.: The diffraction of a blast II. Proc. R. Soc. Lond. 200A, 554–565 (1950)
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)
Liu, T.-P.: Admissible Solutions of Hyperbolic Conservation Laws. Memoirs of the American Mathematical Society, 30(240), 1–78 (1981)
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)
Mach, E.: Über den verlauf von funkenwellenin der ebene und im raume. Sitzungsber. Akad. Wiss. Wien 78, 819–838 (1878)
Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984)
Menikoff, R., Plohr, B.: Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61, 75–130 (1989)
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)
Morawetz, C.S.: Potential theory for regular and Mach reflection of a shock at a wedge. Comm. Pure Appl. Math. 47, 593–624 (1994)
Prandtl, L.: Allgemeine Überlegungen über die Strömung zusammendrückbarer Fluüssigkeiten. Z. Angew. Math. Mech. 16, 129–142 (1938)
Riemann, B.: Über die Fortpflanzung ebener Luftvellen von endlicher Schwingungsweite. Gött. Abh. Math. Cl. 8, 43–65 (1860)
Schulz-Rinne, C.W.: Classification of the Riemann problem for two-dimensional gas dynamics. SIAM J. Math. Anal. 24, 76–88 (1993)
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)
Serre, D.: Shock reflection in gas dynamics. In: Handbook of Mathematical Fluid Dynamics, vol. 4, pp. 39–122. Elsevier, North-Holland (2007)
Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Acta Numer. 29, 701–762 (2020)
Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, New York (1982)
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)
Van Dyke, M.: An Album of Fluid Motion. The Parabolic Press, Stanford (1982)
Von Neumann, J.: Theory of shock waves, Progress Report. U.S. Dept. Comm. Off. Tech. Serv. No. PB32719, Washington, DC (1943)
Von Neumann, J.: Oblique reflection of shocks, Explo. Res. Rep. 12. Navy Department, Bureau of Ordnance, Washington, DC (1943)
Von Neumann, J.: Refraction, intersection, and reflection of shock waves, NAVORD Rep. 203-45. Navy Department, Bureau of Ordnance, Washington, DC (1945)
Von Neumann, J.: Collected Works, vol. 6. Pergamon, New York (1963)
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)
Wagner, D.H.: The Riemann problem in two space dimensions for a single conservation laws. SIAM J. Math. Anal. 14, 534–559 (1983)
Wendroff, B.: The Riemann problem for materials with nonconvex equations of state: I. Isentropic flow. J. Math. Anal. Appl. 38, 454–466 (1972)
Wendroff, B.: The Riemann problem for materials with nonconvex equations of state: II. General flow. J. Math. Anal. Appl. 38, 640–658 (1972)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comp. Phys. 54, 115–173 (1984)
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)
Zhang, T., Zheng, Y.: Two-dimensional Riemann problem for a scalar conservation law. Trans. Am. Math. Soc. 312, 589–619 (1989)
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)
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)
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)
Zheng, Y.: Two-dimensional regular shock reflection for the pressure gradient system of conservation laws. Acta Math. Appl. Sin. 22(2), 177–210 (2006)
Zheng, Y.: Systems of Conservation Laws: Two-Dimensional Riemann Problems, vol. 38. Springer, New York (2012)
Acknowledgements
This paper is dedicated to Professor Tong Zhang (Tung Chang) on the occasion of his 90th birthday, who has been one of the pioneers and main contributors in the analysis of the 2-D Riemann problems; see for example [8,9,10,11,12, 59, 80, 92,93,94] and the references cited therein. The research of Gui-Qiang G. Chen was supported in part by the UK Engineering and Physical Sciences Research Council Awards EP/L015811/1, EP/V008854/1, EP/V051121/1, and the Royal Society-Wolfson Research Merit Award WM090014.
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Chen, GQ.G. Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems. Commun. Appl. Math. Comput. 5, 1015–1052 (2023). https://doi.org/10.1007/s42967-022-00210-4
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DOI: https://doi.org/10.1007/s42967-022-00210-4
Keywords
- 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