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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For two classes of hyperbolic systems of conservation laws *new definitions of a **-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 -shocks. The *weak asymptotics method* developed by the authors is used to describe the propagation of -shock waves to the three types of systems of conservation laws and derive the corresponding Rankine-Hugoniot conditions for -shocks.

**1.**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. MR**1323183 (96e:76107)****2.**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. MR**1756560 (2001f:35244)****3.**V. G. Danilov, G. A. Omelyanov, 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. MR**1995392 (2004f:35021)****4.**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. MR**1882927****5.**V. G. Danilov and V. M. Shelkovich,*Propagation and interaction of shock waves of quasilinear equation.*Nonlinear Studies**8**(2001), no. 1, 135-169. MR**1856223 (2002k:35199)****6.**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. MR**2053197 (2005a:35191)****7.**V. G. Danilov, V. M. Shelkovich,*Dynamics of propagation and interaction of -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)**8.**V. G. Danilov, V. M. Shelkovich,*Propagation and interaction of -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. MR**2088475****9.**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. MR**1384139 (98a:82077)****10.**G. Ercole,*Delta-shock waves as self-similar viscosity limits.*Quart. Appl. Math.**58**(2000), no. 1, 177-199. MR**1739044 (2000j:35187)****11.**Jiaxin Hu,*The Riemann problem for pressureless fluid dynamics with distribution solutions in Colombeau's sense.*Comm. Math. Phys.**194**(1998), 191-205. MR**1628318 (99e:35130)****12.**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. MR**1690829 (2000c:35147)****13.**K. T. Joseph,*A Riemann problem whose viscosity solutions contain -measures.*Asymptotic Analysis**7**(1993), 105-120. MR**1225441 (94f:35083)****14.**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. MR**1330835 (96b:35138)****15.**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. MR**1074190 (91g:35171)****16.**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. MR**1717235 (2000f:76114)****17.**Tai-Ping Liu, Zhouping Xin,*Overcompressive shock waves*, Nonlinear evolution equations that change type, Springer-Verlag, 1990, 139-145. MR**1074191 (91f:35171)****18.**A. Majda,*Compressible fluid flow and systems of conservation laws in several space variables*, Springer-Verlag, New York, 1984. MR**0748308 (85e:35077)****19.**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. MR**1365258 (97b:46052)****20.**M. Nedeljkov,*Delta and singular delta locus for one-dimensional systems of conservation laws.*Math. Meth. Appl. Sci.**27**(2004) 931-955. MR**2055283****21.**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. MR**0694243 (85f:35127)****22.**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. MR**0989562 (89m:85008)****23.**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. MR**1347378 (96g:46025)****24.**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. MR**2014201 (2004i:35220)****25.**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. MR**1466909 (99g:35109)****26.**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. MR**1287550 (95g:35124)****27.**A. I. Volpert,*The space and quasilinear equations.*Math. USSR Sb.**2**(1967), 225-267.**28.**Hanchun Yang,*Riemann problems for a class of coupled hyperbolic systems of conservation laws.*Journal of Differential Equations**159**(1999), 447-484. MR**1730728 (2000j:35184)****29.**Ya. B. Zeldovich,*Gravitationnal instability: An approximate theory for large density perturbations.*Astron. Astrophys.**5**(1970), 84-89.

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC (2000):
35L65,
35L67,
76L05

Retrieve articles in all journals with MSC (2000): 35L65, 35L67, 76L05

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

DOI:
https://doi.org/10.1090/S0033-569X-05-00961-8

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)

Article copyright:
© Copyright 2005
Brown University