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Multidimensional Conservation Laws: Overview, Problems, and Perspective

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Book cover Nonlinear Conservation Laws and Applications

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 153))

Abstract

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.

AMS(MOS) subject classifications. Primary: 35-02, 35L65, 35L67, 35L10, 35F50, 35M10, 35M30, 76H05, 76J20, 76L05, 76G25, 76N15, 76N10, 57R40, 53C42, 74B20, 26B12; Secondary: 35L80, 35L60, 57R42.

The work of Gui-Qiang G. Chen was supported in part by NSF grants DMS-0935967, DMS-0807551, the Royal Society–Wolfson Research Merit Award (UK), and the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1).

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References

  1. S. Alinhac, Existence of rarefaction waves for multidimensional hyperbolic quasilinear systems, Comm. Partial Diff. Eqs. 14 (1989), pp. 173–230.

    MathSciNet  MATH  Google Scholar 

  2. H.W. Alt and L.A. Caffarelli, Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), pp. 105–144.

    MathSciNet  MATH  Google Scholar 

  3. H.W. Alt, L.A. Caffarelli, and A. Friedman, A free-boundary problem for quasilinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), pp. 1–44.

    MathSciNet  MATH  Google Scholar 

  4. L. Ambrosio and C. De Lellis, Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions, Int. Math. Res. Not. 2003 (2003), pp. 2205–2220.

    MATH  Google Scholar 

  5. L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press: New York, 2000.

    MATH  Google Scholar 

  6. G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 (1983), pp. 293–318.

    MathSciNet  MATH  Google Scholar 

  7. M. Artola and A. Majda, Nonlinear development of instabilities in supersonic vortex sheets, I: the basic kink modes, Phys. D. 28 (1987), pp. 253–281; II: Resonant interaction among kink modes, SIAM J. Appl. Math. 49 (1989), pp. 1310–1349; Nonlinear kink modes for supersonic vortex sheets, Phys. Fluids A, 1 (1989), pp. 583–596.

    MathSciNet  MATH  Google Scholar 

  8. M.-J. Bae, G.-Q. Chen, and M. Feldman, Regularity of solutions to regular shock reflection for potential flow, Invent. Math. 175 (2009), pp. 505–543.

    MathSciNet  MATH  Google Scholar 

  9. M.-J. Bae and M. Feldman, Transonic shocks in multidimensional divergent nozzles, Preprint, 2010.

    Google Scholar 

  10. C. Baiocchi and A. Capelo, Variational and Quasi-Variational Inequalities, Applications to Free-Boundary Problems, Vols. 1, 2, John Wiley: New York, 1984.

    Google Scholar 

  11. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 63 (1977), pp. 337–403.

    MATH  Google Scholar 

  12. J.M. Ball, A version of the fundamental theorem for Young measures, In: PDEs and Continuum Models of Phase Transitions (Nice, 1988), Lecture Notes in Phys. 344 (1989), pp. 207–215, Springer-Verlag: Berlin-New York.

    Google Scholar 

  13. G. Ben-Dor, Shock Wave Reflection Phenomena, 2nd Edition, Springer-Verlag: New York, 2007.

    MATH  Google Scholar 

  14. S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations. First-Order Systems and Applications, The Clarendon Press and Oxford University Press: Oxford, 2007.

    Google Scholar 

  15. L. Bers, Existence and uniqueness of subsonic flows past a given profile, Comm. Pure Appl. Math. 7 (1954), pp. 441–504.

    MathSciNet  MATH  Google Scholar 

  16. L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley & Sons, Inc.: New York; Chapman & Hall, Ltd.: London 1958.

    Google Scholar 

  17. H. Berestycki, L. Caffarelli, and L. Nirenberg, Uniform estimates for regularization of free boundary problems, In: Analysis and Partial Differential Equations, pp. 567–619, Dekker: New York, 1990.

    Google Scholar 

  18. S. Bianchini and A. Bressan, Vanshing viscosity solutions of nonlinear hyperbolic systems, Ann. Math. (2), 161 (2005), 223–342.

    MathSciNet  MATH  Google Scholar 

  19. W. Bleakney and A.H. Taub, Interaction of shock waves, Rev. Modern Phys. 21 (1949), pp. 584–605.

    MathSciNet  MATH  Google Scholar 

  20. A. Blokhin, and Y. Trakhinin, Stability of strong discontinuities in fluids and MHD, In: Handbook of Mathematical Fluid Dynamics, Vol. 1, pp. 545–652, Elsevier Science B. V.: Amsterdam, The Netherlands, 2002.

    Google Scholar 

  21. G. Boillat, La Propagation des Ondes, Gauthier-Villars: Paris, 1965.

    MATH  Google Scholar 

  22. G. Boillat, Chocs caractéristiques, C. R. Acad. Sci. Paris Sér. A-B, 274A (1972), pp. 1018–1021; Involutions des systèmes conservatifs, C. R. Acad. Sci. Paris, Sèrie I, 307 (1988), pp. 891–894.

    MathSciNet  Google Scholar 

  23. F. Bouchut, On zero pressure gas dynamics, In: Advances in Kinetic Theory and Computing, pp. 171–190, World Sci. Publishing: River Edge, NJ, 1994.

    Google Scholar 

  24. Y. Brenier, Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Ration. Mech. Anal. 172 (2004), pp. 65–91.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  26. P. Brenner, The Cauchy problem for the symmetric hyperbolic systems in LP, Math. Scand. 19 (1966), pp. 27–37.

    MathSciNet  MATH  Google Scholar 

  27. A. Bressan, Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem, Oxford University Press: Oxford, 2000.

    MATH  Google Scholar 

  28. A. Bressan, An ill-posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova, 110 (2003), pp. 103–117.

    MathSciNet  MATH  Google Scholar 

  29. F. Brezzi and M. Fortin, Mixed and Hydrid Finite Element Methods, Springer- Verlag: New York, 1991.

    Google Scholar 

  30. L.A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries, I. Lipschitz free boundaries are C1,α, Rev. Mat. Iberoamericana, 3 (1987), pp. 139–162; II. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42 (1989), pp. 55–78; III. Existence theory, compactness, and dependence on X, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 15 (1989), pp. 583–602.

    MathSciNet  MATH  Google Scholar 

  31. S. Canić, B.L. Keyfitz, and G. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm. Pure Appl. Math. 53 (2000), pp. 484–511.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  33. S. Canić, B.L. Keyfitz, and E.H. Kim, Free boundary problems for a quasilinear degenerate elliptic equation: regular reflection of weak shocks, Comm. Pure Appl. Math. 55 (2002), pp. 71–92.

    MathSciNet  MATH  Google Scholar 

  34. T. Chang and G.-Q. Chen, Diffraction of planar shock along the compressive corner, Acta Math. Sci. 6 (1986), pp. 241–257.

    MathSciNet  MATH  Google Scholar 

  35. T. Chang, G.-Q. Chen, and S. Yang, On the Riemann Problem for twodimensional Euler equations I: Interaction of shocks and rarefaction waves; II: Interaction of contact discontinuities, Discrete Contin. Dynam. Systems, 1 (1995), pp. 555–584; 6 (2000), pp. 419–430.

    MathSciNet  MATH  Google Scholar 

  36. T. Chang and L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, Longman Scientific & Technical, Harlow; and John Wiley & Sons, Inc.: New York, 1989.

    MATH  Google Scholar 

  37. J.-Y. Chemin, Dynamique des gaz à masse totale finie, Asymptotic Anal. 3 (1990), pp. 215–220.

    MathSciNet  MATH  Google Scholar 

  38. G.-Q. Chen, Hyperbolic systems of conservation laws with a symmetry, Commun. Partial Diff. Eqs. 16 (1991), pp. 1461–1487.

    MATH  Google Scholar 

  39. G.-Q. Chen Euler Equations and Related Hyperbolic Conservation Laws, In: Handbook of Differential Equations: Evolutionary Differential Equations, Vol. 2, pp. 1–104, 2005, Elsevier Science B.V: Amsterdam, The Netherlands.

    Google Scholar 

  40. G.-Q. Chen, Remarks on global solutions to the compressible Euler equations with spherical symmetry, Proc. Royal Soc. Edinburgh, 127A (1997), pp. 243–259.

    Google Scholar 

  41. G.-Q. Chen, Vacuum states and stability of rarefaction waves for compressible flow, Meth. Appl. Anal. 7 (2000), pp. 337–362.

    MATH  Google Scholar 

  42. G.-Q. Chen and J. Chen, Vacuum states and global stability of rarefaction waves for the Euler equations for compressible fluids, J. Hyper. Diff. Eqns. 4 (2007), pp. 105–122.

    MATH  Google Scholar 

  43. G.-Q. Chen, J. Chen, and M. Feldman, Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles, J. Math. Pures Appl. 88 (2007), pp. 191–218.

    MathSciNet  MATH  Google Scholar 

  44. G.-Q. Chen, J. Chen, and M. Feldman, Existence and stability of transonic shocks in the full Euler flows past wedges, Preprint, November 2010.

    Google Scholar 

  45. G.-Q. Chen, C.M. Dafermos, M. Slemrod, and D. Wang, On two-dimensional sonic-subsonic flow, Commun. Math. Phys. 271 (2007), pp. 635–647.

    MathSciNet  MATH  Google Scholar 

  46. G.-Q. Chen and B. Fang, Stability of transonic shock-fronts in steady potential flow past a perturbed cone, Discrete Conti. Dynamical Systems, 23 (2009), pp. 85–114.

    MathSciNet  MATH  Google Scholar 

  47. G.-Q. Chen and M. Feldman, Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc. 16 (2003), pp. 461–494; Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations, Comm. Pure Appl. Math. 57 (2004), pp. 310–356; Free boundary problems and transonic shocks for the Euler equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 3 (2004), pp. 827–869.

    MathSciNet  MATH  Google Scholar 

  48. G.-Q. Chen and M. Feldman, Existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007), pp. 185–242

    MathSciNet  MATH  Google Scholar 

  49. G.-Q. Chen and M. Feldman, Potential theory for shock reflection by a largeangle wedge, Proceedings of the National Academy of Sciences USA (PNAS), 102 (2005), pp. 15368–15372; Global solutions to shock reflection by largeangle wedges for potential flow, Annals of Math. 171 (2010), 1067–1182.

    MathSciNet  MATH  Google Scholar 

  50. G.-Q. Chen and M. Feldman, Shock reflection-diffraction and multidimensional conservation laws, In: Hyperbolic Problems: Theory, Numerics, and Applications, pp. 25–51, Proc. Sympos. Appl. Math. 67, AMS: Providence, RI, 2010.

    Google Scholar 

  51. G.-Q. Chen and M. Feldman, Shock reflection-diffraction and nonlinear partial differential equations of mixed type, Contemporary Mathematics, AMS: Providence (to appear); Potential Theory for Shock Reflection-Diffraction and von Neumann’s Conjectures, Preprint, July 2010.

    Google Scholar 

  52. G.-Q. Chen and H. Frid, Decay of entropy solutions of nonlinear conservation laws, Arch. Ration. Mech. Anal. 146 (1999), pp. 95–127; Large-time behavior of entropy solutions of conservation laws, J. Diff. Eqs. 152 (1999), pp. 308–357.

    MathSciNet  MATH  Google Scholar 

  53. G.-Q. Chen and H. Frid, Large-time behavior of entropy solutions in L∞ for multidimensional conservation laws, In: Advances in Nonlinear PDE and Related Areas, pp. 28–44, World Scientific, Singapore, 1998.

    Google Scholar 

  54. G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 (1999), pp. 89–118; Extended divergence-measure fields and the Euler equations for gas dynamics, Commun. Math. Phys. 236 (2003), pp. 251–280.

    MathSciNet  MATH  Google Scholar 

  55. G.-Q. Chen, H. Frid, and Y. Li, Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics, Commun. Math. Phys. 228 (2002), pp. 201–217.

    MathSciNet  MATH  Google Scholar 

  56. G.-Q. Chen and P.-T. Kan, Hyperbolic conservation laws with umbilic degeneracy (I)–(II): Arch. Ration. Mech. Anal. 130 (1995), pp. 231–276; 160 (2001), pp. 325–354.

    MathSciNet  MATH  Google Scholar 

  57. G.-Q. Chen and P. LeFloch, Compressible Euler equations with general pressure law, Arch. Ration. Mech. Anal. 153 (2000), pp. 221–259; Existence theory for the isentropic Euler equations, Arch. Ration. Mech. Anal. 166 (2003), pp. 81–98.

    MathSciNet  MATH  Google Scholar 

  58. G.-Q. Chen and T.-H. Li Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge, J. Diff. Eqs. 244 (2008), pp. 1521–1550.

    MATH  Google Scholar 

  59. G.-Q. Chen and H. Liu, Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal. 34 (2003), pp. 925–938; Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Phys. D, 189 (2004), pp. 141–165.

    MathSciNet  MATH  Google Scholar 

  60. G.-Q. Chen and M. Rascle, Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws, Arch. Ration. Mech. Anal. 153 (2000), pp. 205–220.

    MathSciNet  MATH  Google Scholar 

  61. G.-Q. Chen, M. Slemrod, and D. Wang, Vanishing viscosity method for transonic flow, Arch. Ration. Mech. Anal. 189 (2008), pp. 159–188.

    MathSciNet  MATH  Google Scholar 

  62. G.-Q. Chen, M. Slemrod, and D. Wang, Isometric immersion and compensated compactness, Commun. Math. Phys. 294 (2010), pp. 411–437.

    MathSciNet  MATH  Google Scholar 

  63. G.-Q. Chen, M. Slemrod, and D. Wang, Weak continuity of the Gauss-Codazzi- Ricci equations for isometric embedding, Proc. Amer. Math. Soc. 138 (2010), pp. 1843–1852.

    MathSciNet  MATH  Google Scholar 

  64. G.-Q. Chen and M. Torres, Divergence-measure fields, sets of finite perimeter, and conservation laws, Arch. Ration. Mech. Anal. 175 (2005), pp. 245–267.

    MathSciNet  MATH  Google Scholar 

  65. G.-Q. Chen and M. Torres, On entropy solutions without bounded variation for hyperbolic systems of conservation laws, Preprint, 2010.

    Google Scholar 

  66. G.-Q. Chen, M. Torres, and W. Ziemer, Gauss-Green theorem for weakly differentiable fields, sets of finite perimeter, and balance laws, Comm. Pure Appl. Math. 62 (2009), pp. 242–304; Measure-theoretical analysis and nonlinear conservation laws, Pure Appl. Math. Quarterly, 3 (2007), pp. 841–879.

    MathSciNet  MATH  Google Scholar 

  67. G.-Q. Chen, D.-H. Wang, and X.-Z. Yang, Evolution of discontinuity and formation of triple-shock pattern in solutions to a two-dimensional hyperbolic system of conservation laws, SIAM J. Math. Anal. 41 (2009), 1–25.

    MathSciNet  Google Scholar 

  68. G.-Q. Chen and Y.-G. Wang, Existence and stability of compressible currentvortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal. 187 (2007), pp. 369–408; Analysis of compressible currentvortex sheets in three-dimensional magnetohydrodynamics, Preprint, November 2010.

    Google Scholar 

  69. G.-Q. Chen, Y. Zhang, and D. Zhu, Existence and stability of supersonic Euler flows past Lipschitz wedges, Arch. Ration. Mech. Anal. 181 (2006), pp. 261–310.

    MathSciNet  MATH  Google Scholar 

  70. G.-Q. Chen, Y. Zhang, and D. Zhu, Existence and stability of supersonic vortex sheets in Euler flows past Lipschitz walls, SIAM J. Math. Anal. 38 (2007), pp. 1660–1693.

    MathSciNet  MATH  Google Scholar 

  71. S.-X. Chen, Existence of stationary supersonic flows past a point body, Arch. Ration. Mech. Anal. 156 (2001), pp. 141–181.

    MathSciNet  MATH  Google Scholar 

  72. S.-X. Chen, Stability of transonic shock fronts in two-dimensional Euler systems, Trans. Amer. Math. Soc. 357 (2005), pp. 287–308; Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems, Trans. Amer. Math. Soc. 360 (2008), pp. 5265–5289.

    MathSciNet  MATH  Google Scholar 

  73. S.-X. Chen, Compressible flow and transonic shock in a diverging nozzle, Commun. Math. Phys. 289 (2009), pp. 75–106.

    MATH  Google Scholar 

  74. S.-X. Chen Stability of a Mach configuration, Comm. Pure Appl. Math. 59 (2006), pp. 1–35; Mach configuration in pseudo-stationary compressible flow, J. Amer. Math. Soc. 21 (2008), pp. 63–100.

    MathSciNet  MATH  Google Scholar 

  75. S.-X. Chen and B. Fang, Stability of transonic shocks in supersonic flow past a wedge, J. Diff. Eqs. 233 (2007), pp. 105–135.

    MathSciNet  MATH  Google Scholar 

  76. S.-X. Chen and H. Yuan, Transonic shocks in compressible flow passing a duct for three-dimensional Euler systems, Arch. Ration. Mech. Anal. 187 (2008), pp. 523–556.

    MathSciNet  MATH  Google Scholar 

  77. S.-X. Chen, Z. Xin, and H. Yin, Global shock waves for the supersonic flow past a perturbed cone, Commun. Math. Phys. 228 (2002), pp. 47–84.

    MathSciNet  MATH  Google Scholar 

  78. A.J. Chorin, Vorticity and Turbulence, Springer-Verlag: New York, 1994.

    MATH  Google Scholar 

  79. D. Christodoulou, The Formation of Shocks in 3-Dimensional Fluids, European Mathematical Society (EMS): Zürich, 2007.

    MATH  Google Scholar 

  80. D. Christodoulou, The Formation of Black Holes in General Relativity, European Mathematical Society (EMS): Zürich, 2009.

    MATH  Google Scholar 

  81. J.D. Cole and L.P. Cook, Transonic Aerodynamics, Elsevier: Amsterdam, 1986.

    MATH  Google Scholar 

  82. P. Constantin, Some mathematical problems of fluid mechanics, Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zürich, 1994), pp. 1086–1095, Birkhäuser: Basel, 1995.

    Google Scholar 

  83. J.F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J. 53 (2004), 941–1012; Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. (4), 41 (2008), 85–39; Uniqueness of 2-D compressible vortex sheets, Commun. Pure Appl. Anal. 8 (2009), 1439–1450.

    MathSciNet  MATH  Google Scholar 

  84. R. Courant and K.O. Friedrichs, Supersonic Flow and Shock Waves, Springer- Verlag: New York, 1962.

    Google Scholar 

  85. R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience Publishers, Inc.: New York, 1953.

    Google Scholar 

  86. M.S. Cramer and A.R. Seebass, Focusing of a weak shock at an arâte, J. Fluid Mech. 88 (1978), pp. 209–222.

    MATH  Google Scholar 

  87. D. Cui and H. Yin, Global supersonic conic shock wave for the steady supersonic flow past a cone: polytropic gas, J. Diff. Eqs. 246 (2009), pp. 641–669.

    MathSciNet  MATH  Google Scholar 

  88. B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity of Nonlinear Functionals, Lecture Notes in Math. 922, Springer-Verlag: New York, 1982.

    Google Scholar 

  89. C.M. Dafermos, Quasilinear hyperbolic systems with involutions, Arch. Ration. Mech. Anal. 94 (1986), 373–389.

    MathSciNet  MATH  Google Scholar 

  90. C.M. Dafermos, Entropy and the stability of classical solutions of hyperbolic systems of conservation laws, Lecture Notes in Math. 1640, pp. 48–69, Springer- Verlag: Berlin, 1996.

    Google Scholar 

  91. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag: Berlin, 2010.

    MATH  Google Scholar 

  92. C. De Lellis, F. Otto, and M. Westdickenberg, Structure of entropy solutions for multidimensional scalar conservation laws, Arch. Ration. Mech. Anal. 170 (2003), pp. 137–184.

    MathSciNet  MATH  Google Scholar 

  93. C. De Leliss and L. Székelyhidi Jr. The Euler equations as a differential inclusion, Ann. of Math. (2), 170 (2009), 1417–1436; On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal. 195 (2010), 225–260.

    MathSciNet  Google Scholar 

  94. S. Demoulini, D.M. Stuart, and A.E. Tzavaras, A variational approximation scheme for three-dimensional elastodynamics with polyconvex energy, Arch. Ration. Mech. Anal. 157 (2001), pp. 325–344.

    MathSciNet  MATH  Google Scholar 

  95. R.J. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292 (1985), pp. 383–420.

    MathSciNet  Google Scholar 

  96. R.J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Ration. Mech. Anal. 88 (1985), pp. 223–270; Convergence of viscosity method for isentropic gas dynamics, Commun. Math. Phys. 91 (1983), pp. 1–30.

    MathSciNet  MATH  Google Scholar 

  97. R.J. DiPerna, Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal. 88 (1985), 223–270.

    MathSciNet  MATH  Google Scholar 

  98. G. Dong, Nonlinear Partial Differential Equations of Second Order, Transl. Math. Monographs, 95, AMS: Providence, RI, 1991.

    MATH  Google Scholar 

  99. W. E. Y. Rykov and Y. Sinai, Generalized variational principles, global weak solutions, and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Commun. Math. Phys. 177 (1996), pp. 349–380.

    MathSciNet  MATH  Google Scholar 

  100. V. Elling and T.-P. Liu, The ellipticity principle for self-similar potential flows, J. Hyperbolic Differential Equations, 2 (2005), 909–917.

    MathSciNet  MATH  Google Scholar 

  101. V. Elling and T.-P. Liu, Supersonic flow onto a solid wedge, Comm. Pure Appl. Math. 61 (2008), pp. 1347–1448.

    MathSciNet  MATH  Google Scholar 

  102. B. Engquist and W. E, Large time behavior and homogenization of solutions of two-dimensional conservation laws, Comm. Pure Appl. Math. 46 (1993), pp. 1–26.

    MathSciNet  MATH  Google Scholar 

  103. L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS 72, AMS: Providence, RI, 1990.

    MATH  Google Scholar 

  104. B. Fang, Stability of transonic shocks for the full Euler system in supersonic flow past a wedge, Math. Methods Appl. Sci. 29 (2006), pp. 1–26.

    MathSciNet  Google Scholar 

  105. H. Federer, Geometric Measure Theory, Springer-Verlag: Berlin-Heidelberg- New York, 1969.

    MATH  Google Scholar 

  106. R. Finn and D. Gilbarg, Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations, Acta Math. 98 (1957), pp. 265–296.

    MathSciNet  MATH  Google Scholar 

  107. S. Friedland, J.W. Robbin, and J. Sylvester, On the crossing rule, Comm. Pure Appl. Math. 37 (1984), pp. 19–38.

    MathSciNet  MATH  Google Scholar 

  108. K.O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), pp. 345–392.

    MathSciNet  MATH  Google Scholar 

  109. K. Friedrichs and P. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci., U.S.A. 68 (1971), pp. 1686–1688.

    MathSciNet  MATH  Google Scholar 

  110. I.M. Gamba and C.S. Morawetz, A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow: existence theorem for potential flow, Comm. Pure Appl. Math. 49 (1996), pp. 999–1049.

    MathSciNet  MATH  Google Scholar 

  111. W. Gangbo and M. Westdickenberg, Optimal transport for the system of isentropic Euler equations, Comm. Partial Diff. Eqs. 34 (2009), 1041–1073.

    MathSciNet  MATH  Google Scholar 

  112. E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag: Basel, 1984.

    MATH  Google Scholar 

  113. J. Glimm, Solutions in the large for nonlinear hyperbolic system of equations, Comm. Pure Appl. Math. 18 (1965), pp. 95–105.

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  115. J. Glimm, C. Klingenberg, O. McBryan, B. Plohr, D. Sharp, and S. Yaniv, Front tracking and two-dimensional Riemann problems, Adv. Appl. Math. 6 (1985), pp. 259–290.

    MathSciNet  MATH  Google Scholar 

  116. J. Glimm and A. Majda, Multidimensional Hyperbolic Problems and Computations, IMA Volumes in Mathematics and its Applications, 29, Springer- Verlag: New York, 1991.

    MATH  Google Scholar 

  117. S. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR. 139 (1961), pp. 521–523, English transl: Soviet Math. 2 (1961), pp. 947–949; Elements of Continuum Mechanics, Moscow: Nauka, 1978; Lois de conservation et integrals d’énergie des équations hyperboliques, Lecture Notes in Math. 1270 (1987), pp. 135–149, Berlin: Springer.

    MathSciNet  Google Scholar 

  118. E. Grenier, Existence globale pour le système des gaz sans pression, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), pp. 171–174.

    MathSciNet  MATH  Google Scholar 

  119. C.-H. Gu, A method for solving the supersonic flow past a curved wedge (in Chinese), Fudan Univ. J. 7 (1962), pp. 11–14.

    Google Scholar 

  120. O. Guès, G. Métiver, M. Williams, and K. Zumbrun, Navier-Stokes regularization of multidimensional Euler shocks, Ann. Sci. École Norm. Sup. (4) 39 (2006), pp. 75–175.

    MATH  Google Scholar 

  121. K.G. Guderley, The Theory of Transonic Flow, Pergamon Press: Oxford- London-Paris-Frankfurt; Addison-Wesley Publishing Co. Inc.: Reading, Mass. 1962.

    MATH  Google Scholar 

  122. L. Guo, W. Sheng, and T. Zhang, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal. 9 (2010), pp. 431–458.

    MathSciNet  MATH  Google Scholar 

  123. J. Hadamard, Leçons sur la Propagation des Ondes et les Équations de l’Hydrodynamique, Hermann, Paris, 1903 (Reprinted by Chelsea 1949).

    MATH  Google Scholar 

  124. Q. Han and J.-X. Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, AMS: Providence, RI, 2006.

    MATH  Google Scholar 

  125. D. Hoff, The zero-Mach limit of compressible flows, Comm. Math. Phys. 192 (1998), pp. 543–554.

    MathSciNet  MATH  Google Scholar 

  126. H. Holden and N.H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag: New York, 2002.

    MATH  Google Scholar 

  127. F. Huang and Z. Wang, Well posedness for pressureless flow, Commun. Math. Phys. 222 (2001), pp. 117–146.

    MATH  Google Scholar 

  128. J. Hunter, Transverse diffraction of nonlinear waves and singular rays, SIAM J. Appl.Math. 48 (1988), pp. 1–37; Nonlinear wave diffraction, In: Geometrical Optics and Related Topics (Cortona, 1996), pp. 221–243, Birkhäuser Boston: Boston, MA, 1997.

    MathSciNet  MATH  Google Scholar 

  129. J. Hunter and J. Keller, Weak shock diffraction, Wave Motion, 6 (1984), pp. 79–89 and pp. 321.

    MathSciNet  MATH  Google Scholar 

  130. H.K. Jenssen, Blowup for systems of conservation laws, SIAM J. Math. Anal. 31 (2000), pp. 894–908.

    MathSciNet  MATH  Google Scholar 

  131. T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal. 58 (1975), pp. 181–205; Quasilinear equations of evolution with applications to partial differential equations, Lecture Notes in Math. 448, pp. 25–70, Springer-Verlag: Berlin, 1975.

    MATH  Google Scholar 

  132. J.B. Keller and A. Blank, Diffraction and reflection of pulses by wedges and corners, Comm. Pure Appl. Math. 4 (1951), pp. 75–94.

    MathSciNet  MATH  Google Scholar 

  133. J.B. Keller and L. Ting, Periodic vibrations of systems governed by nonlinear partial differential equations, Comm. Pure Appl. Math. 19 (1966), pp. 371– 420.

    MathSciNet  MATH  Google Scholar 

  134. A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Methods Partial Diff. Eqs. 18 (2002), pp. 584–608.

    MathSciNet  MATH  Google Scholar 

  135. A.G. Kuz’min, Boundary Value Problems for Transonic Flow, John Wiley & Sons, 2002.

    Google Scholar 

  136. P.D. Lax, Shock waves and entropy, In: Contributions to Nonlinear Functional Analysis, E. A. Zarantonello (ed.), pp. 603–634, Academic Press: New York, 1971.

    Google Scholar 

  137. P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM: Philadelphia, 1973.

    MATH  Google Scholar 

  138. P.D. Lax, The multiplicity of eigenvalues, Bull. Amer. Math. Soc. (N.S.), 6 (1982), pp. 213–214.

    MathSciNet  MATH  Google Scholar 

  139. P.D. Lax, Hyperbolic systems of conservation laws in several space variables, In: Current Topics in PDEs, pp. 327–341, Kinokuniya: Tokyo, 1986.

    Google Scholar 

  140. P.D. Lax and X.-D. Liu, Solutions of two dimensional Riemann problem of gas dynamics by positive schemes, SIAM J. Sci. Comput. 19 (1998), pp. 319–340.

    MathSciNet  MATH  Google Scholar 

  141. P.G. LeFloch, Hyperbolic Systems of Conservation Laws: The theory of classical and nonclassical shock waves, Birkhäuser: Zürich, 2002.

    MATH  Google Scholar 

  142. J. Li, Note on the compressible Euler equations with zero temperature, Appl. Math. Lett. 14 (2001), pp. 519–523.

    MathSciNet  MATH  Google Scholar 

  143. J. Li, T. Zhang, and S. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Longman (Pitman Monographs 98): Essex, 1998.

    Google Scholar 

  144. J. Li and Y. Zheng, Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations, Comm. Math. Phys. 296 (2010), pp. 303–321.

    MathSciNet  MATH  Google Scholar 

  145. J. Li, Z. Xin, and H. Yin, On transonic shocks in a nozzle with variable end pressure, Commun. Math. Phys. 291 (2009), pp. 111–150.

    MathSciNet  MATH  Google Scholar 

  146. T.-T. Li, On a free boundary problem, Chinese Ann. Math. 1 (1980), pp. 351–358.

    MATH  Google Scholar 

  147. T.-T. Li and W.-C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Press: Durham, 1985.

    MATH  Google Scholar 

  148. M.J. Lighthill, The diffraction of a blast I, Proc. Royal Soc. London, 198 (1949), pp. 454–47; The diffraction of a blast II, Proc. Roy. Soc. London, 200A (1950), pp. 554–565.

    MathSciNet  Google Scholar 

  149. W.-C. Lien and T.-P. Liu, Nonlinear stability of a self-similar 3-dimensional gas flow, Commun. Math. Phys. 204 (1999), pp. 525–549.

    MathSciNet  MATH  Google Scholar 

  150. P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vols. 1–2, Oxford University Press: Oxford, 1998.

    MATH  Google Scholar 

  151. P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9), 77 (1998), pp. 585–627.

    MathSciNet  MATH  Google Scholar 

  152. P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), pp. 169–191.

    MathSciNet  MATH  Google Scholar 

  153. L. Liu and H.R. Yuan, Stability of cylindrical transonic shocks for twodimensional steadt compressible Euler system, J. Hyper. Diff. Eqs. 5 (2008), pp. 347–379.

    MathSciNet  MATH  Google Scholar 

  154. T.-P. Liu and T. Yang, L1 stability of weak solutions for 2 × 2 systems of hyperbolic conservation laws, J. Amer. Math. Soc. 12 (1999), pp. 729–774; Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 52 (1999), pp. 1553–1586.

    MathSciNet  MATH  Google Scholar 

  155. E. Mach, Über den verlauf von funkenwellenin der ebene und im raume, Sitzungsber. Akad. Wiss. Wien, 78 (1878), pp. 819–838.

    Google Scholar 

  156. A. Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc. 275, AMS: Providence, 1983; The existence of multidimensional shock fronts, Mem. Amer. Math. Soc. 281, AMS: Providence, 1983.

    Google Scholar 

  157. A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag: New York, 1984.

    MATH  Google Scholar 

  158. A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press: Cambridge, 2002.

    MATH  Google Scholar 

  159. A. Majda and E. Thomann, Multidimensional shock fronts for second order wave equations, Comm. Partial Diff. Eqs. 12 (1987), pp. 777–828.

    MathSciNet  MATH  Google Scholar 

  160. T. Makino, S. Ukai, and S. Kawashima, Sur la solution à support compact de l’iquations d’Euler compressible, Japan J. Appl. Math. 3 (1986), pp. 249–257.

    MathSciNet  MATH  Google Scholar 

  161. V.G. Maz’ya and B.A. Plamenevskivı, Estimates in LP and in Hölder classes and the Miranda-Agman maximun principle for solutions of elliptic boundary value problems in domains with sigular points on the boundary, Amer. Math. Soc. Transl., 123 (1984), pp. 1–56.

    Google Scholar 

  162. G. Métivier, Stability of multi-dimensional weak shocks, Comm. Partial Diff. Eqs. 15 (1990), pp. 983–1028.

    MATH  Google Scholar 

  163. C.S. Morawetz, On a weak solution for a transonic flow problem, Comm. Pure Appl. Math. 38 (1985), pp. 797–818. On steady transonic flow by compensated compactness, Meth. Appl. Anal. 2 (1995), pp. 257–268.

    MathSciNet  MATH  Google Scholar 

  164. C.S. Morawetz, Mixed equations and transonic flow, J. Hyper. Diff. Eqns. 1 (2004), pp. 1–26.

    MathSciNet  MATH  Google Scholar 

  165. C.S. Morawetz, Potential theory for regular and Mach reflection of a shock at a wedge, Comm. Pure Appl. Math. 47 (1994), pp. 593–624.

    MathSciNet  MATH  Google Scholar 

  166. C.S. Morawetz, On the non-existence of continuous transonic flows past profiles I,II,III, Comm. Pure Appl. Math. 9 (1956), pp. 45–68; 10 (1957), pp. 107– 131; 11 (1958), pp. 129–144.

    MathSciNet  MATH  Google Scholar 

  167. C.B. Morrey, Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), pp. 25–53.

    MathSciNet  MATH  Google Scholar 

  168. F. Murat, The injection of the positive cone of H-1 in W-1, q is completely continuous for all q < 2, J. Math. Pures Appl. (9), 60 (1981), pp. 309–322.

    MathSciNet  MATH  Google Scholar 

  169. F. Murat, Compacité par compensation, Ann. Suola Norm. Pisa (4), 5 (1978), pp. 489–507; Compacité par compensation, condition nécessaire et suffisante de continuité faible sous une hypothèse de range constant, Ann. Scuola Norm. Sup. Pisa (4), 8 (1981), pp. 69–102; A survey on compensated compactness, In: Contributions to Modern Calculus of Variations (Bologna, 1985), pp. 145–183, Longman Sci. Tech.: Harlow, 1987.

    MathSciNet  MATH  Google Scholar 

  170. J. von Neumann, Oblique reflection of shocks, Navy Department, Bureau of Ordance, Explosive Research Report, No. 12, 1943.

    Google Scholar 

  171. J. von Neumann, Collect Works, Vol. 5, Pergamon: New York, 1963.

    Google Scholar 

  172. S. Osher, M. Hafez, and W. Whitlow, Entropy condition satisfying approximations for the full potential equation of transonic flow, Math. Comp. 44 (1985), pp. 1–29.

    MathSciNet  MATH  Google Scholar 

  173. E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyper. Diff. Eqs. 4 (2007), pp. 729–770.

    MathSciNet  MATH  Google Scholar 

  174. B. Perthame, Time decays: an analogy between kinetic transport, Schrödinger and gas dynamics equations, In: Nonlinear Partial Differential Equations and Their Applications, pp. 281–293, Longman: Harlow, 1998.

    Google Scholar 

  175. B. Perthame, Kinetic Formations of Conservation Laws, Oxford University Press: Oxford, 2002.

    Google Scholar 

  176. T. Qin, Symmetrizing the nonlinear elastodynamic system, J. Elasticity, 50 (1998), pp. 245–252.

    MathSciNet  MATH  Google Scholar 

  177. J. Rauch, BV estimates fail for most quasilinear hyperbolic systems in dimension great than one, Commun. Math. Phys. 106 (1986), pp. 481–484.

    MathSciNet  MATH  Google Scholar 

  178. J.-F. Rodrigues, Obstacle Problems in Mathematical Physics, Elsevier Science Publishers B.V. 1987.

    Google Scholar 

  179. T. Ruggeri and A. Strumia, Main field and convex covariant density for quasilinear hyperbolic systems, Ann. Inst. Henri Poincaré, Section A, 34 (1981), pp. 65–84.

    MathSciNet  MATH  Google Scholar 

  180. D.G. Schaeffer, Supersonic flow past a nearly straight wedge, Duke Math. J. 43 (1976), pp. 637–670.

    MathSciNet  MATH  Google Scholar 

  181. D. Schaeffer and M. Shearer, The classification of 2×2 systems of nonstrictly hyperbolic conservation laws, with application to oil recovery, Comm. Pure Appl. Math. 40 (1987), pp. 141–178; Riemann problems for nonstrictly hyperbolic 2 × 2 systems of conservation laws, Trans. Amer. Math. Soc. 304 (1987), pp. 267–306.

    MathSciNet  MATH  Google Scholar 

  182. C.W. Schulz-Rinne, J.P. Collins, and H.M. Glaz, Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. Sci. Comput. 14 (1993), pp. 1394–1414.

    MathSciNet  MATH  Google Scholar 

  183. D. Serre, Écoulements de fluides parfaits en deux variables indépendantes de type espace. Rèflexion d’un choc plan par un di`edre compressif (French), Arch. Ration. Mech. Anal. 132 (1995), pp. 15–36.

    MathSciNet  MATH  Google Scholar 

  184. D. Serre, Spiral waves for the two-dimensional Riemann problem for a compressible fluid, Ann. Fac. Sci. Toulouse Math. (6), 5 (1996), pp. 125–135.

    MathSciNet  MATH  Google Scholar 

  185. D. Serre, Hyperbolicity of the nonlinear models of Maxwell’s equations, Arch. Ration. Mech. Anal. 172 (2004), pp. 309–331.

    MathSciNet  MATH  Google Scholar 

  186. D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Ration. Mech. Anal. 191 (2009), pp. 539–577.

    MathSciNet  MATH  Google Scholar 

  187. W.-C. Sheng and G. Yin, Transonic shock and supersonic shock in the regular reflection of a planar shock, Z. Angew. Math. Phys. 60 (2009), pp. 438–449.

    MathSciNet  MATH  Google Scholar 

  188. W.-C. Sheng and T. Zhang, The Riemann Problem for the Transportation Equations in Gas Dynamics, Mem. Amer.Math. Soc. 654, AMS: Providence, 1999.

    Google Scholar 

  189. M. Shiffman, On the existence of subsonic flows of a compressible fluid, J. Ration. Mech. Anal. 1 (1952), pp. 605–652.

    MathSciNet  Google Scholar 

  190. T.C. Sideris, Formation of singularities in three-dimensional compressible fluids, Commun. Math. Phys. 101 (1985), pp. 475–485.

    MathSciNet  MATH  Google Scholar 

  191. B. Skews and J. Ashworth, The physical nature of weak shock wave reflection, J. Fluid Mech. 542 (2005), pp. 105–114.

    MathSciNet  MATH  Google Scholar 

  192. E. Tabak and R. Rosales, Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection, Phys. Fluids, 6 (1994), pp. 1874–1892.

    MathSciNet  MATH  Google Scholar 

  193. E. Tadmor, M. Rascle, and P. Bagnerini, Compensated compactness for 2D conservation laws, J. Hype. Diff. Eqs. 2 (2005), 697–712.

    MathSciNet  MATH  Google Scholar 

  194. D. Tan and T. Zhang, Two dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws I: four-J cases, J. Diff. Eqs. 111 (1994), pp. 203–254; II: Initial data involving some rarefaction waves, J. Diff. Eqs. 111 (1994), pp. 255–283.

    MathSciNet  MATH  Google Scholar 

  195. L. Tartar, Compensated compactness and applications to partial differential equations. In: Res. Notes in Math. 39, pp. 136–212, Pitman: Boston-London, 1979.

    Google Scholar 

  196. L. Tartar, Une nouvelle méthode de resolution d’equations aux derivées partielles nonlinéaires, Lecture Notes in Math. 665, pp. 228–241, Springer- Verlag: Berlin, 1977; The compensated compactness method applied to systems of conservation laws, In: Systems of Nonlinear P.D.E., J.M. Ball (eds.), pp. 263–285, NATO Series, C. Reidel publishing Co. 1983.

    Google Scholar 

  197. A. Tesdall and J.K. Hunter, Self-similar solutions for weak shock reflection, SIAM J. Appl. Math. 63 (2002), pp. 42–61.

    MathSciNet  MATH  Google Scholar 

  198. A.M. Tesdall, R. Sanders, and B. Keyfitz, Self-similar solutions for the triple point paradox in gasdynamics, SIAM J. Appl. Math. 68 (2008), pp. 1360–1377.

    MathSciNet  MATH  Google Scholar 

  199. R. Timman, Unsteady motion in transonic flow, In: Symposium Transsonicum (IUTAM, Aachen, Sept. 1962), pp. 394–401, ed. K. Oswatitsch, Springer- Verlag: Berlin, 1964.

    Google Scholar 

  200. Y. Trakhinin, Existence of compressible current-vortex sheets: variable coefficients linear analysis, Arch. Ration. Mech. Anal. 177 (2005), 331–366; The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal. 191 (2009), 245–310.

    MathSciNet  MATH  Google Scholar 

  201. M. Van Dyke, An Album of Fluid Motion, The Parabolic Press: Stanford, 1982.

    Google Scholar 

  202. A. Vasseur, Strong traces for solutions to multidimensional scalar conservation laws, Arch. Ration. Mech. Anal. 160 (2001), pp. 181–193.

    MathSciNet  MATH  Google Scholar 

  203. A. Volpert, The Space BV and quasilinear equations, Mat. Sb. (N.S.) 73 (1967), pp. 255–302; Math. USSR Sbornik, 2 (1967), pp. 225–267 (in English).

    MathSciNet  Google Scholar 

  204. Z. Wang, F. Huang, and X. Ding, On the Cauchy problem of transportation equations, Acta Math. Appl. Sinica (English Ser.), 13 (1997), pp. 113–122.

    MathSciNet  MATH  Google Scholar 

  205. G.B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience: New York, 1973.

    Google Scholar 

  206. Z. Xin, W. Yan, and H. Yin, Transonic shock problem for the Euler system in a nozzle, Arch. Ration. Mech. Anal. 194 (2009), 1–47.

    MathSciNet  MATH  Google Scholar 

  207. S.-T. Yau, Review of geometry and analysis, In: Mathematics: Frontiers and Perspectives, pp. 353–401, International Mathematics Union, Eds. V. Arnold, M. Atiyah, P. Lax, and B. Mazur, AMS: Providence, 2000.

    Google Scholar 

  208. G.I. Zabolotskaya and R.V. Khokhlov, Quasi-plane waves in the nonlinear acoustics of confined beams, Sov. Phys.-Acoustics, 15 (1969), pp. 35–40.

    Google Scholar 

  209. A.R. Zakharian, M. Brio, J.K. Hunter, and G.M. Webb, The von Neumann paradox in weak shock reflection, J. Fluid Mech. 422 (2000), pp. 193–205.

    MathSciNet  MATH  Google Scholar 

  210. Ya.B. Zeldovich, Gravitational instability: an approximate theory for large density perturbations, Astronaut. Astrophys. 5 (1970), pp. 84–89.

    Google Scholar 

  211. T. Zhang and Y. Zheng, Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics, SIAM J. Math. Anal. 21 (1990), pp. 593–630.

    MathSciNet  MATH  Google Scholar 

  212. Y.-Q. Zhang, Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary, SIAM J. Math. Anal. 31 (1999), pp. 166–183; Steady supersonic flow past an almost straight wedge with large vertex angle, J. Diff. Eqs. 192 (2003), pp. 1–46.

    MathSciNet  MATH  Google Scholar 

  213. Y. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems, Birkhäuser: Boston, 2001.

    MATH  Google Scholar 

  214. Y. Zheng, A global solution to a two-dimensional Riemann problem involving shocks as free boundaries, Acta Math. Appl. Sin. Engl. Ser. 19 (2003), pp. 559–572.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  216. W.P. Ziemer, Cauchy flux and sets of finite perimeter, Arch. Ration. Mech. Anal. 84 (1983), pp. 189–201.

    MathSciNet  MATH  Google Scholar 

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  1. Oxford Centre for Nonlinear PDE, Mathematical Institute, University of Oxford, Oxford, OX1, 3LB, UK

    Gui-Qiang G. Chen

  2. Department of Mathematics, Northwestern University, Evanston, IL, 60208, USA

    Gui-Qiang G. Chen

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    Gui-Qiang G. Chen

  3. , School of Mathematics, University of Minnesota, Church Street S.E, 206, Minneapolis, 55455, Minnesota, USA

    Marta Lewicka

  4. , Department of Mathematics, University of Pittsburgh, Pittsburgh, 15260, Pennsylvania, USA

    Dehua Wang

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Chen, GQ.G. (2011). Multidimensional Conservation Laws: Overview, Problems, and Perspective. In: Bressan, A., Chen, GQ., Lewicka, M., Wang, D. (eds) Nonlinear Conservation Laws and Applications. The IMA Volumes in Mathematics and its Applications, vol 153. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9554-4_2

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