$\delta$- and $\delta'$-shock wave types of singular solutions of systems of conservation laws and transport and concentration processes

V. M. Shelkovich

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 63 (2008) no. 3, pp. 473-546

Résumé

This is a survey of some results and problems connected with the theory of generalized solutions of quasi-linear conservation law systems which can admit delta-shaped singularities. They are the so-called $\delta$-shock wave type solutions and the recently introduced $\delta^{(n)}$-shock wave type solutions, $n=1,2,\dots$, which cannot be included in the classical Lax–Glimm theory. The case of $\delta$- and $\delta'$-shock waves is analyzed in detail. A specific analytical technique is developed to deal with such solutions. In order to define them, some special integral identities are introduced which extend the concept of weak solution, and the Rankine–Hugoniot conditions are derived. Solutions of Cauchy problems are constructed for some typical systems of conservation laws. Also investigated are multidimensional systems of conservation laws (in particular, zero-pressure gas dynamics systems) which admit $\delta$-shock wave type solutions. A geometric aspect of such solutions is considered: they are connected with transport and concentration processes, and the balance laws of transport of ‘volume’ and ‘area’ to $\delta$- and $\delta'$-shock fronts are derived for them. For a ‘zero-pressure gas dynamics’ system these laws are the mass and momentum transport laws. An algebraic aspect of these solutions is also considered: flux-functions are constructed for them which, being non-linear, are nevertheless uniquely defined Schwartz distributions. Thus, a singular solution of the Cauchy problem generates algebraic relations between its components (distributions).

Bibliographie

[1] D. J. Korchinski, Solution of a Riemann problem for $2\times2$ systems of conservation laws possesing no classical weak solution, Thesis, Adelphi Univ., New York, 1977

[2] Ya. B. Zeldovich, A. D. Myshkis, Elementy matematicheskoi fiziki, Nauka, M., 1973 | MR

[3] A. N. Kraiko, “On discontinuity surfaces in a medium devoid of ‘proper’ pressure”, J. Appl. Math. Mech., 43:3 (1979), 539–549 | DOI | MR | Zbl

[4] E. Yu. Panov, V. M. Shelkovich, “$\delta'$-shock waves as a new type of solutions to systems of conservation laws”, J. Differential Equations, 228:1 (2006), 49–86 | DOI | MR | Zbl

[5] E. Yu. Panov, V. M. Shelkovich, “$\delta'$-shock waves as a new type of singular solutions to hyperbolic systems of conservation laws”, Doklady Math., 73:2 (2006), 264–268 | DOI | MR

[6] E. Yu. Panov, V. M. Shelkovich, “Definition of $\delta'$-shock waves type solutions and the Rankine–Hugoniot conditions”, Proceedings of the International Conference “Days on Diffraction” (St. Petersburg, 2005), 2005, 187–199

[7] V. M. Shelkovich, “The Riemann problem admitting $\delta$-, $\delta'$-shocks, and vacuum states (the vanishing viscosity approach)”, J. Differential Equations, 231:2 (2006), 459–500 | DOI | MR | Zbl

[8] L. C. Evans, Partial differential equations, Grad. Stud. Math., 19, Amer. Math. Soc., Providence, RI, 1998 | MR | Zbl

[9] S. Albeverio, V. M. Shelkovich, “On the delta-shock front problem”, Analytical Approaches to Multidimensional Balance Laws, vol. 2, Nova Science Publishers, Inc., 2005, 45–88

[10] F. Bouchut, “On zero pressure gas dynamics”, Advances in Kinetic Theory and Computing, Ser. Adv. Math. Appl. Sci., 22, World Scientific, River Edge, NJ, 1994, 171–190 | MR | Zbl

[11] Gui-Qiang Chen, Hailiang 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., 34:4 (2003), 925–938 | DOI | MR | Zbl

[12] Gui-Qiang Chena, Hailiang Liu, “Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids”, Phys. D, 189:1–2 (2004), 141–165 | DOI | MR | Zbl

[13] S. Chiang, “On the interaction of $\delta$-shocks and contact vacuum states”, Z. Angew. Math. Phys., 57:6 (2006), 940–959 | DOI | MR | Zbl

[14] V. G. Danilov, V. M. Shelkovich, “Propagation and interaction of delta-shock waves of a hyperbolic system of conservation laws”, Hyperbolic Problems: Theory, Numerics, Applications. Proceedings of the Ninth International Conference on Hyperbolic Problems held in CalTech (Pasadena, 2002), Springer-Verlag, Berlin, 2003, 483–492 | MR | Zbl

[15] V. G. Danilov, V. M. Shelkovich, “Propagation and interaction of $\delta$-shock waves for hyperbolic systems of conservation laws”, Dokl. Math., 69:1 (2004), 4–8 | MR | Zbl

[16] V. G. Danilov, V. M. Shelkovich, “Delta-shock wave type solution of hyperbolic systems of conservation laws”, Quart. Appl. Math., 63:3 (2005), 401–427 | MR

[17] V. G. Danilov, V. M. Shelkovich, “Dynamics of propagation and interaction of $\delta$-shock waves in conservation law systems”, J. Differential Equations, 211:2 (2005), 333–381 | DOI | MR | Zbl

[18] W. E, Yu. G. Rykov, Ya. G. Sinai,, “Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics”, Comm. Math. Phys., 177:2 (1996), 349–380 | DOI | MR | Zbl

[19] G. Ercole, “Delta-shock waves as self-similar viscosity limits”, Quart. Appl. Math., 58:1 (2000), 177–199 | MR | Zbl

[20] F. Huang, “Existence and uniqueness of discontinuous solutions for a class of nonstrictly hyperbolic systems”, Advances in Nonlinear Partial Differential Equations and Related Areas (Beijing, 1997), World Scientific, River Edge, NJ, 1998, 187–208 | MR | Zbl

[21] Feiming Huang, Zhen Wang, “Well posedness for pressureless flow”, Comm. Math. Phys., 222:1 (2001), 117–146 | DOI | MR | Zbl

[22] B. L. Keyfitz, “Conservation laws, delta-shocks ans singular shocks”, Nonlinear Theory of Generalized Functions (Vienna, 1997), Chapman/CRC Res. Notes Math., 401, Chapman/CRC, Boca Raton, FL, 1999, 99–111 | MR | Zbl

[23] B. L. Keyfitz, H. C. Kranzer, “Spaces of weighted measures for conservation laws with singular shock solutions”, J. Differential Equations, 118:2 (1995), 420–451 | DOI | MR | Zbl

[24] B. L. Keyfitz, R. Sanders, M. Sever, “Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow”, Discrete Contin. Dyn. Syst. Ser. B, 3:4 (2003), 541–563 | DOI | MR | Zbl

[25] B. L. Keyfitz, M. Sever, Fu Zhang, “Viscous singular shock structure for a nonhyperbolic two-fluid model”, Nonlinearity, 17:5 (2004), 1731–1747 | DOI | MR | Zbl

[26] H. C. Kranzer, B. L. Keyfitz, “A strictly hyperbolic system of conservation laws admitting singular shocks”, Nonlinear Evolution Equations That Change Type, IMA Vol. Math. Appl., 27, Springer, New York, 1990, 107–125 | MR | Zbl

[27] P. Le Floch, “An existence and uniqueness result for two nonstrictly hyperbolic systems”, Nonlinear Evolution Equations That Change Type, IMA Vol. Math. Appl., 27, Springer, New York, 1990 | MR | Zbl

[28] M. Sever, Distribution solutions of nonlinear systems of conservation laws, Mem. Amer. Math. Soc., No 190, 2007 | MR | Zbl

[29] V. M. Shelkovich, “Delta-shock waves of a class of hyperbolic systems of conservation laws”, Patterns and Waves (St. Petersburg, July 8–13, 2002), AkademPrint, St. Petersburg, 2003, 155–168 | MR

[30] V. M. Shelkovich, A specific hyperbolic system of conservation laws admitting delta-shock wave type solutions, Preprint No 2003-059, NTNU, Trondheim, 2003

[31] V. M. Shelkovich, “Delta-shocks, the Rankine–Hugoniot conditions, and singular superposition of distributions”, Proceedings of the International Conference “Days on Diffraction” (St. Petersburg, 2004), 2004, 175–196

[32] Wancheg Sheng, Tong Zhang, The Riemann problem for the transportaion equations in gas dynamics, Mem. Amer. Math. Soc., No. 137, 1999 | MR | Zbl

[33] De Chun Tan, Tong Chang, Yu Xi Zheng, “Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws”, J. Differential Equations, 112:1 (1994), 1–32 | DOI | MR | Zbl

[34] Hanchun Yang, “Riemann problems for a class of coupled hyperbolic systems of conservation laws”, J. Differential Equations, 159:2 (1999), 447–484 | DOI | MR | Zbl

[35] J. Glimm, “Solutions in the large for nonlinear hyperbolic systems of equations”, Comm. Pure Appl. Math., 18:4 (1965), 697–715 | DOI | MR | Zbl

[36] P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS-NSF Regional Conf. Ser. Appl. Math., 11, SIAM, Philadelphia, PA, 1973 | MR | Zbl

[37] Jiequan Li, Tong Zhang, “On the initial-value problem for zero-pressure gas dynamics”, Hyperbolic Problems: Theory, Numerics, Applications (Zurich, 1998), vol. II, Internat. Ser. Numer. Math., 130, Birkhäuser, Basel, 1999, 629–640 | MR | Zbl

[38] Jiequan Li, Wei Li, “Riemann problem for the zero-pressure flow in gas dynamics”, Progr. Natur. Sci. (English Ed.), 11:5 (2001), 331–344 | MR

[39] Jiequan Li, Hanchun Yang, “Delta-shocks as limit of vanishing viscosity for multidimensional zero-pressure gas dynamics”, Quart. Appl. Math., 59:2 (2001), 315–342 | MR | Zbl

[40] Yu. G. Rykov, “The propagation of shock waves in 2-D system of pressureless gas dynamics”, Hyperbolic Problems: Theory, Numerics, Applications (Zürich, 1998), vol. II, Internat. Ser. Numer. Math., 130, Birkhäuser, Basel, 1999, 813–822 | MR | Zbl

[41] Yu. G. Rykov, “On the non-Hamiltonian character of shocks in 2-D pressureless gas”, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 5:1 (2002), 55–78 | MR | Zbl

[42] M. Sever, “An existence theorem in the large for zero-pressure gas dynamics”, Differential Integral Equations, 14:9 (2001), 1077–1092 | MR | Zbl

[43] Hanchun Yang, “Generalized plane delta-shock waves for $n$-dimensional zero-pressure gas dynamics”, J. Math. Anal. Appl., 260:1 (2001), 18–35 | DOI | MR | Zbl

[44] A. I. Vol'pert, “The spaces $BV$ and quasilinear equations”, Math. USSR-Sb., 2:2 (1967), 225–267 | DOI | MR | Zbl

[45] G. Dal Maso, P. G. Le Floch, F. Murat, “Definition and weak stability of nonconservative products”, J. Math. Pures Appl. (9), 74:6 (1995), 483–548 | MR | Zbl

[46] P. Le Floch, “Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form”, Comm. Partial Differential Equations, 13:6 (1988), 669–727 | DOI | MR | Zbl

[47] J.-F. Colombeau, Elementary introduction to new generalized functions, North-Holland Math. Stud. 113, Notes on Pure Mathematics, 103, North-Holland, Amsterdam, 1985 | MR | Zbl

[48] J.-F. Colombeau, “Multiplication of distributions”, Bull. Amer. Math. Soc., 23:2 (1990), 251–268 | DOI | MR | Zbl

[49] S. Schecter, “Existence of Dafermos profiles for singular shocks”, J. Differential Equations, 205:1 (2004), 185–210 | DOI | MR | Zbl

[50] D. G. Schaeffer, S. Schecter, M. Shearer, “Nonstrictly hyperbolic conservation laws with a parabolic line”, J. Differential Equations, 103:1 (1993), 94–126 | DOI | MR | Zbl

[51] M. Sever, “Viscous structure of singular shocks”, Nonlinearity, 15:3 (2002), 705–725 | DOI | MR | Zbl

[52] V. M. Shelkovich, “On delta-shocks and singular shocks”, Hyperbolic Problems. Theory, Numerics and Applications. Proceedings of the 11th International Conference on Hyperbolic Problems (Lyon, 2006), eds. S. Benzoni-Gavage et al., Springer, Berlin, 2008, 971–979 | Zbl

[53] R. J. DiPerna, “Measure-valued solutions to conservation laws”, Arch. Ration Mech. Anal., 88:3 (1985), 223–270 | DOI | MR | Zbl

[54] M. Grosser, M. Kunzinger, M. Oberguggenberger, R. Steinbauer, Geometric theory of generalized functions with applications to general relativity, Math. Appl., 537, Kluwer Acad. Publ., Dordrecht, 2001 | MR | Zbl

[55] M. Oberguggenberger, Multiplication of distributions and applications to partial differential equations, Pitman Res. Notes Math. Ser., 259, Longman, Harlow; Wiley, New York, 1992 | MR | Zbl

[56] M. Oberguggenberger, “Generalizef functions in nonlinear models – a survey”, Proceedings of the Third World Congress of Nonlinear Analysts, Part 8 (Catania, 2000), Nonlinear Anal., 47:8 (2001), 5029–5040 | DOI | MR | Zbl

[57] Jiaxin Hu, “The Riemann problem for pressureless fluid dynamics with distributional solutions in Colombeau's sense”, Comm. Math. Phys., 194:1 (1998), 191–205 | DOI | MR | Zbl

[58] K. T. Joseph, “Explicit generalized solutions to a system of conservation laws”, Proc. Indian Acad. Sci. Math. Sci., 109:4 (1999), 401–409 | DOI | MR | Zbl

[59] L. Schwartz, “Sur l'impossibilité de la multiplication des distributions”, C. R. Acad. Sci. Paris, 239 (1954), 847–848 | MR | Zbl

[60] V. M. Shelkovich, “New versions of the Colombeau algebras”, Math. Nachr., 278:11 (2005), 1318–1340 | DOI | MR | Zbl

[61] V. M. Shelkovich, “The Colombeau algebra and an algebra of asymptotical distributions”, Proceedings of the International Conference on Generalized Functions (University of French West Indies, 2000), Cambridge Scientific Publishers, 2004, 317–328

[62] V. M. Shelkovich, “Colombeau generalized functions: A theory based on harmonic regularizations”, Math. Notes, 63:2 (1998), 275–279 | DOI | MR | Zbl

[63] V. G. Danilov, V. M. Shelkovich, “Propagation and interaction of nonlinear waves to quasilinear equations”, Hyperbolic Problems: Theory, Numerics, Applications, vol. I (Magdeburg, 2000), Internat. Ser. Numer. Math., 140, Birkhäuser, Basel, 2001, 267–276 | MR

[64] V. G. Danilov, V. M. Shelkovich, “Propagation and interaction of shock waves of quasilinear equation”, Nonlinear Stud., 8:1 (2001), 135–169 | MR | Zbl

[65] V. G. Danilov, G. A. Omel'yanov, V. M. Shelkovich, “Weak asymptotics method and interaction of nonlinear waves”, Asymptotic methods for wave and quantum problems, Amer. Math. Soc. Transl. Ser. 2, 208, Amer. Math. Soc., Providence, RI, 2003, 33–165 | MR | Zbl

[66] M. G. García-Alvarado, R. Flores-Espinoza, G. A. Omel'yanov, “Interaction of shock waves in gas dynamics: uniform in time asymptotics”, Int. J. Math. Math. Sci., 19 (2005), 3111–3126 | DOI | MR | Zbl

[67] R. Flores-Espinoza, G. A. Omel'yanov, “Asymptotic behavior for the centered-rarefaction appearence problem”, Electron. J. Differ. Equ., 2005, no. 148, 1–25 | MR | Zbl

[68] V. G. Danilov, D. Mitrovic, “Weak asymptotics of shock wave formation process”, Nonlinear Anal., 61:4 (2005), 613–635 | DOI | MR | Zbl

[69] V. G. Danilov, D. Mitrovic, Delta shock wave formation in the case of triangular hyperbolic system of conservation laws, Preprint No 2006-057, NTNU, Trondheim, 2006; ; Z. Differential Equation (to appear) arXiv: math/0612763

[70] V. G. Danilov, “On singularities of continuity equation solutions”, Nonlinear Anal., 68:6 (2008), 1640–1651 | DOI | MR | Zbl

[71] A. Majda, “The stability of multidimensional shock fronts”, Mem. Amer. Math. Soc., 41:275 (1983) | MR | Zbl

[72] A. Majda, “The existence of multidimensional shock fronts”, Mem. Amer. Math. Soc., 43:281 (1983) | MR | Zbl

[73] A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Appl. Math. Sci., 53, Springer-Verlag, New York, 1984 | MR | Zbl

[74] Tai-Ping Liu, Zhou Ping Xin, “Overcompressive shock waves”, Nonlinear Evolution Equations That Change Type, IMA Vol. Math. Appl., 27, Springer, New York, 1990, 139–145 | MR | Zbl

[75] S. F. Shandarin, Ya. B. Zeldovich, “The large-scale structure of the universe: turbulence, intermittence, structures in self-gravitating medium”, Rev. Modern Phys., 61:2 (1989), 185–220 | DOI | MR

[76] Ya. B. Zeldovich, “Gravitational instability: An approximate theory for large density perturbations”, Astronom. Astrophys., 5 (1970), 84–89

[77] S. Albeverio, S. A. Molchanov, D. Surgailis, “Stratified structure of the Universe and the Burgers' equation – a probabilistic approach”, Probab. Theory Related Fields, 100:4 (1994), 457–484 | DOI | MR | Zbl

[78] A. N. Kraiko, S. M. Sulaimanova, “Two-fluid flows of a mixture of a gas and solid particles with ‘films’ and ‘filaments’ appearing in flows past impermeable surfaces”, J. Appl. Math. Mech., 47:4 (1983), 507–516 | DOI | Zbl

[79] A. N. Kraiko, “The two-fluid model of flow of gas with particles dispersed in it”, J. Appl. Math. Mech., 46:1 (1983), 75–82 | DOI | Zbl

[80] A. N. Osiptsov, “Investigation of regions of unbounded growth of the particle concentration in disperse flows”, Fluid Dynam., 19 (1984), 378–385 | DOI | Zbl

[81] A. N. Osiptsov, “Modified Lagrangian method for calculating the particle concentration in dusty-gas flows with intersecting particle trajectories”, Proceedings of the 3 International Conference on Multiphase Flows (Lyon, 1998), 1998

[82] A. N. Osiptsov, “Lagrangian modeling of dust admixture in gas flows”, Astrophys. Space Sci. Lib., 274:1–2 (2000), 377–386 | DOI | Zbl

[83] F. Berthelin, P. Degond, M. Delitala, M. Rascle, “A model for the formation and evolution of traffic jams”, Arch. Ration. Mech. Anal., 187:2 (2008), 185–220 | DOI | MR | Zbl

[84] M. Mazzotti, “Local equilibrium theory for the binary chromatography of species subject to a generalized Langmuir isotherm”, Ind. Eng. Chem. Res., 45:15 (2006), 5332–5350 | DOI

[85] M. Mazzotti, “Occurrence of a delta-shock in non-linear chromatography”, 6th International Congress on Industrial and Applied Mathematics (Zurich, 2007), 2007

[86] I. Fouxon, B. Meerson, M. Assaf, E. Livne, “Formation and evolution of density singularities in hydrodynamics of inelastic gases”, Phys. Rev. E, 75:5 (2007), 050301(R) | DOI

[87] V. M. Shelkovich, “Validity and naturalness of definitions of $\delta$- and $\delta'$-shock type solutions to systems of conservation laws”, Proceedings of the International Conference “Days on Diffraction” (St. Petersburg, 2006), 2006, 266–279

[88] A. Yu. Khrennikov, V. M. Shelkovich, “On the algebraic aspect of singular solutions to conservation laws systems”, Mathematical Modelling of Wave Phenomena: 2nd Conference on Mathematical Modelling of Wave Phenomena (Växjö, Sweden, 2005), AIP Conference Proceedings, 834, no. 1, 2006, 206–213 | DOI

[89] F. Poupaud, “Global smooth solutions of some quasi-linear hyperbolic systems with large data”, Ann. Fac. Sci. Toulouse Math. (6), 8:4 (1999), 649–659 | MR | Zbl

[90] R. P. Kanwal, Generalized Functions: Theory and technique, Birkhäuser, Boston, MA, 1998 | MR | Zbl

[91] I. M. Gel'fand, G. E. Shilov, Generalized functions. I: Properties and operations, Academic Press, New York–London, 1964 | MR | MR | Zbl | Zbl

[92] V. M. Shelkovich, “The Rankine–Hugoniot conditions and balance laws for $\delta$-shock waves”, J. Math. Sci., 151:1 (2008), 2781–2792 | DOI | MR

[93] D. E. Betounes, “Dirac tensor distributions for moving submanifolds of $\mathbb{R}^n$”, J. Math. Phys., 24:11 (1983), 2566–2572 | DOI | MR | Zbl

[94] D. E. Betounes, “Nonclassical fields with singularities on the moving surface”, J. Math. Phys., 23:12 (1982), 2304–2311 | DOI | MR

[95] V. S. Vladimirov, Generalized functions in mathematical physics, Mir, Moscow, 1979 | MR | MR | Zbl | Zbl

[96] I. S. Aranson, L. S. Tsimring, “Patterns and collective behavior in granular media: Theoretical concepts”, Rev. Modern Phys., 78:2 (2006), 641–692 | DOI

[97] B. Meerson, “Nonlinear dynamics of radiative condensations in optically thin plasmas”, Rev. Modern Phys., 68:1 (1996), 215–257 | DOI

[98] I. Fouxon, B. Meerson, M. Assaf, E. Livne, “Formation of density singularities in ideal hydrodynamics of freely cooling inelastic gases: a family of exact solutions”, Phys. Fluids, 19 (2007), 093303 ; arXiv: 0704.084 | DOI

[99] D. A. Drew, S. L. Passman, Theory of multicomponent fluids, Appl. Math. Sci., 135, Springer, New York, 1999 | DOI | MR | Zbl