Delta-shock wave type solution of hyperbolic systems of conservation laws
Authors:
V. G. Danilov and V. M. Shelkovich
Journal:
Quart. Appl. Math. 63 (2005), 401-427
MSC (2000):
Primary 35L65; Secondary 35L67, 76L05
DOI:
https://doi.org/10.1090/S0033-569X-05-00961-8
Published electronically:
August 17, 2005
MathSciNet review:
2169026
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Abstract: For two classes of hyperbolic systems of conservation laws new definitions of a $\delta$-shock wave type solution are introduced. These two definitions give natural generalizations of the classical definition of the weak solutions. It is relevant to the notion of $\delta$-shocks. The weak asymptotics method developed by the authors is used to describe the propagation of $\delta$-shock waves to the three types of systems of conservation laws and derive the corresponding Rankine–Hugoniot conditions for $\delta$-shocks.
- F. Bouchut, On zero pressure gas dynamics, Advances in kinetic theory and computing, Ser. Adv. Math. Appl. Sci., vol. 22, World Sci. Publ., River Edge, NJ, 1994, pp. 171–190. MR 1323183
- V. G. Danilov, V. P. Maslov, and V. M. Shelkovich, Algebras of the singularities of singular solutions of first-order quasilinear strictly hyperbolic systems, Teoret. Mat. Fiz. 114 (1998), no. 1, 3–55 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 114 (1998), no. 1, 1–42. MR 1756560, DOI https://doi.org/10.1007/BF02557106
- V. G. Danilov, G. A. Omel′yanov, and V. M. Shelkovich, Weak asymptotics method and interaction of nonlinear waves, Asymptotic methods for wave and quantum problems, Amer. Math. Soc. Transl. Ser. 2, vol. 208, Amer. Math. Soc., Providence, RI, 2003, pp. 33–163. MR 1995392, DOI https://doi.org/10.1090/trans2/208/02
- V. G. Danilov and V. M. Shelkovich, Propagation and interaction of nonlinear waves to quasilinear equations, Hyperbolic problems: theory, numerics, applications, Vol. I, II (Magdeburg, 2000) Internat. Ser. Numer. Math., 140, vol. 141, Birkhäuser, Basel, 2001, pp. 267–276. MR 1882927
- V. G. Danilov and V. M. Shelkovich, Propagation and interaction of shock waves of quasilinear equation, Nonlinear Stud. 8 (2001), no. 1, 135–169. MR 1856223
- V. G. Danilov and V. M. Shelkovich, Propagation and interaction of delta-shock waves of a hyperbolic system of conservation laws, Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2003, pp. 483–492. MR 2053197
D-S4 V. G. Danilov, V. M. Shelkovich, Dynamics of propagation and interaction of $\delta$-shock waves in hyperbolic systems. pp.40, Preprint 2003-068 at the url: http://www.math.ntnu.no/conservation/2003/068.html (to appear in the Journal of Differential Equations)
- V. G. Danilov and V. M. Shelkovich, Propagation and interaction of $\delta $-shock waves of hyperbolic systems of conservation laws, Dokl. Akad. Nauk 394 (2004), no. 1, 10–14 (Russian). MR 2088475
- Weinan E, Yu. G. Rykov, and 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 (1996), no. 2, 349–380. MR 1384139
- Grey Ercole, Delta-shock waves as self-similar viscosity limits, Quart. Appl. Math. 58 (2000), no. 1, 177–199. MR 1739044, DOI https://doi.org/10.1090/qam/1739044
- Jiaxin Hu, The Riemann problem for pressureless fluid dynamics with distribution solutions in Colombeau’s sense, Comm. Math. Phys. 194 (1998), no. 1, 191–205. MR 1628318, DOI https://doi.org/10.1007/s002200050355
- Feimin 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 Sci. Publ., River Edge, NJ, 1998, pp. 187–208. MR 1690829
- K. T. Joseph, A Riemann problem whose viscosity solutions contain $\delta $-measures, Asymptotic Anal. 7 (1993), no. 2, 105–120. MR 1225441
- Barbara Lee Keyfitz and Herbert C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations 118 (1995), no. 2, 420–451. MR 1330835, DOI https://doi.org/10.1006/jdeq.1995.1080
- Philippe LeFloch, An existence and uniqueness result for two nonstrictly hyperbolic systems, Nonlinear evolution equations that change type, IMA Vol. Math. Appl., vol. 27, Springer, New York, 1990, pp. 126–138. MR 1074190, DOI https://doi.org/10.1007/978-1-4613-9049-7_10
- Jiequan Li and Tong Zhang, On the initial-value problem for zero-pressure gas dynamics, Hyperbolic problems: theory, numerics, applications, Vol. II (Zürich, 1998) Internat. Ser. Numer. Math., vol. 130, Birkhäuser, Basel, 1999, pp. 629–640. MR 1717235
- Tai-Ping Liu and Zhou Ping Xin, Overcompressive shock waves, Nonlinear evolution equations that change type, IMA Vol. Math. Appl., vol. 27, Springer, New York, 1990, pp. 139–145. MR 1074191, DOI https://doi.org/10.1007/978-1-4613-9049-7_11
- A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR 748308
- Gianni Dal Maso, Philippe G. Lefloch, and François Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483–548. MR 1365258
- Marko Nedeljkov, Delta and singular delta locus for one-dimensional systems of conservation laws, Math. Methods Appl. Sci. 27 (2004), no. 8, 931–955. MR 2055283, DOI https://doi.org/10.1002/mma.480
- B. L. Roždestvenskiĭ and N. N. Janenko, Systems of quasilinear equations and their applications to gas dynamics, Translations of Mathematical Monographs, vol. 55, American Mathematical Society, Providence, RI, 1983. Translated from the second Russian edition by J. R. Schulenberger. MR 694243
- S. F. Shandarin and Ya. B. Zel′dovich, The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium, Rev. Modern Phys. 61 (1989), no. 2, 185–220. MR 989562, DOI https://doi.org/10.1103/RevModPhys.61.185
- V. M. Shelkovich, An associative-commutative algebra of distributions that includes multipliers, and generalized solutions of nonlinear equations, Mat. Zametki 57 (1995), no. 5, 765–783, 800 (Russian, with Russian summary); English transl., Math. Notes 57 (1995), no. 5-6, 536–549. MR 1347378, DOI https://doi.org/10.1007/BF02304423
- V. M. Shelkovich, Delta-shock waves of a class of hyperbolic systems of conservation laws, Patterns and waves (Saint Petersburg, 2002) AkademPrint, St. Petersburg, 2003, pp. 155–168. MR 2014201
- Wancheng Sheng and Tong Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc. 137 (1999), no. 654, viii+77. MR 1466909, DOI https://doi.org/10.1090/memo/0654
- De Chun Tan, Tong Zhang, and Yu Xi Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112 (1994), no. 1, 1–32. MR 1287550, DOI https://doi.org/10.1006/jdeq.1994.1093
V A. I. Volpert, The space $BV$ and quasilinear equations. Math. USSR Sb. 2 (1967), 225–267.
- Hanchun Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differential Equations 159 (1999), no. 2, 447–484. MR 1730728, DOI https://doi.org/10.1006/jdeq.1999.3629
Z Ya. B. Zeldovich, Gravitationnal instability: An approximate theory for large density perturbations. Astron. Astrophys. 5 (1970), 84–89.
B F. Bouchut, On zero pressure gas dynamics, in “Advances in Kinetic Theory and Computing”, Series on Advances in Mathematics for Applied Sciences, 22, 171–190, World Scientific, River Edge, NJ, 1994.
D-M-S V. G. Danilov, V. P. Maslov, V. M. Shelkovich, Algebra of singularities of singular solutions of first-order quasilinear strictly hyperbolic systems. Theor. Math. Phys. 114 (1998), no. 1, 1–42.
D-O-S V. G. Danilov, G. A. Omel$’$yanov, V. M. Shelkovich, Weak Asymptotics Method and Interaction of Nonlinear Waves, in Mikhail Karasev (ed.), “Asymptotic Methods for Wave and Quantum Problems”, Amer. Math. Soc. Transl., Ser. 2, 208, 2003, 33–163.
D-S1 V. G. Danilov, V. M. Shelkovich, Propagation and interaction of nonlinear waves to quasilinear equations, Hyperbolic problems: Theory, Numerics, Applications (Eighth International Conference in Magdeburg, February/March 2000, v.I). International Series of Numerical Mathematics, v. 140, Birkhäuser, Basel/Switzerland, 2001, 267–276.
D-S2 V. G. Danilov and V. M. Shelkovich, Propagation and interaction of shock waves of quasilinear equation. Nonlinear Studies 8 (2001), no. 1, 135–169.
D-S3 V. G. Danilov, V. M. Shelkovich, Propagation and interaction of delta-shock waves of a hyperbolic system of conservation laws, In Hou, Thomas Y.; Tadmor, Eitan (Eds.), Hyperbolic Problems: Theory, Numerics, Applications. Proceedings of the Ninth International Conference on Hyperbolic Problems held in CalTech, Pasadena, March 25-29, 2002, Springer-Verlag, 2003, 483–492.
D-S4 V. G. Danilov, V. M. Shelkovich, Dynamics of propagation and interaction of $\delta$-shock waves in hyperbolic systems. pp.40, Preprint 2003-068 at the url: http://www.math.ntnu.no/conservation/2003/068.html (to appear in the Journal of Differential Equations)
D-S5 V. G. Danilov, V. M. Shelkovich, Propagation and interaction of $\delta$-shock waves to hyperbolic systems of conservation laws. Dokl. Akad. Nauk 394 (2004), no. 1, 10–14. English transl. in Russian Doklady Mathematics 69 (2004), no. 1.
E-R-S Weinan E., Yu. 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 (1996), 349–380.
E G. Ercole, Delta-shock waves as self-similar viscosity limits. Quart. Appl. Math. 58 (2000), no. 1, 177–199.
Hu Jiaxin Hu, The Riemann problem for pressureless fluid dynamics with distribution solutions in Colombeau’s sense. Comm. Math. Phys. 194 (1998), 191–205.
H Feiming Huang, Existence and uniqueness of discontinuous solutions for a class of nonstrictly hyperbolic systems, In Chen, Gui-Qiang (ed.) et al. Advances in nonlinear partial differential equations and related areas. Proceeding of conf. dedicated to Prof. Xiaqi Ding, China, 1997, 187–208.
J K. T. Joseph, A Riemann problem whose viscosity solutions contain $\delta$-measures. Asymptotic Analysis 7 (1993), 105–120.
Ke-Kr B. Lee Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions. Journal of Differential Equations 118 (1995), 420–451.
LeFl P. Le Floch, An existence and uniqueness result for two nonstrictly hyperbolic systems,25-FEB-2005 Nonlinear Evolution Equations That Change Type, Springer-Verlag, 1990, 126–138.
L-Z J. Li and Tong Zhang, On the initial-value problem for zero-pressure gas dynamics, Hyperbolic problems: Theory, Numerics, Applications. Seventh International Conference in Zürich, February 1998, Birkhäuser, Basel, Boston, Berlin, 1999, 629–640.
Liu-Xin Tai-Ping Liu, Zhouping Xin, Overcompressive shock waves, Nonlinear evolution equations that change type, Springer-Verlag, 1990, 139–145.
Maj A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Springer-Verlag, New York, 1984.
M-LeFl-M G. Dal Maso, P. G. Le Floch, and F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995), 483–548.
N M. Nedeljkov, Delta and singular delta locus for one-dimensional systems of conservation laws. Math. Meth. Appl. Sci. 27 (2004) 931–955.
R-Y B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations, Moscow, Nauka, 1978 (in Russian) B. L. Rozhdestvenskii and N. N. Janenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics, New York, Am. Math., 1983.
Shan-Z S. F. Shandarin and Ya. B. Zeldovich, The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium. Rev. Mod. Phys. 61 (1989), 185–220.
S V. M. Shelkovich, An associative-commutative algebra of distributions that includes multipliers, generalized solutions of nonlinear equations. Mathematical Notices 57 (1995), no. 5, 765–783.
S2 V. M. Shelkovich, Delta-shock waves of a class of hyperbolic systems of conservation laws, in A. Abramian, S. Vakulenko, V. Volpert (Eds.), “Patterns and Waves”, AkademPrint, St. Petersburg, 2003, 155–168.
S-Z Wancheng Sheng, Tong Zhang, The Riemann problem for the transportation equations in gas dynamics. Memoirs of the Amer. Math. Soc., 137, no. 654, (1999), 1–77.
T-Z-Z De Chun Tan, Tong Zhang and Yu Xi Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. Journal of Differential Equations 112 (1994), 1–32.
V A. I. Volpert, The space $BV$ and quasilinear equations. Math. USSR Sb. 2 (1967), 225–267.
Y Hanchun Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws. Journal of Differential Equations 159 (1999), 447–484.
Z Ya. B. Zeldovich, Gravitationnal instability: An approximate theory for large density perturbations. Astron. Astrophys. 5 (1970), 84–89.
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Additional Information
V. G. Danilov
Affiliation:
Department of Mathematics, Moscow Technical University of Communication and Informatics, Aviamotornaya, 8a, 111024, Moscow, Russia
Email:
danilov@miem.edu.ru
V. M. Shelkovich
Affiliation:
Department of Mathematics, St.-Petersburg State Architecture and Civil Engineering University, 2 Krasnoarmeiskaya 4, 190005, St. Petersburg, Russia
Email:
shelkv@vs1567.spb.edu
Keywords:
Hyperbolic systems of conservation laws,
zero-pressure gas dynamics system,
delta-shock wave type solutions,
the Rankine–Hugoniot conditions of delta-shocks,
the weak asymptotics method
Received by editor(s):
August 5, 2003
Published electronically:
August 17, 2005
Additional Notes:
The first author (V. D.) was supported in part by Grant 05-01-00912 of Russian Foundation for Basic Research, SEP-CONACYT Grant 41421, SEP-CONACYT Grant 43208 (Mexico)
The second author (V. S.) was supported in part by DFG Project 436 RUS 113/593/3, Grant 02-01-00483 of Russian Foundation for Basic Research, and SEP-CONACYT Grant 41421, SEP-CONACYT Grant 43208 (Mexico)
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