Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure gas dynamics

Authors:
Jiequan Li and Hanchun Yang

Journal:
Quart. Appl. Math. **59** (2001), 315-342

MSC:
Primary 76N99; Secondary 35B35, 35L65, 35M20, 35Q35

DOI:
https://doi.org/10.1090/qam/1827367

MathSciNet review:
MR1827367

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study zero-pressure gas dynamics, which is a nonstrict hyperbolic system of nonlinear conservation laws with delta-shock waves in solutions. By using the generalized Rankine-Hugoniot relations to solve the Riemann problem with two pieces of constant initial data, multidimensional planar delta-shock waves dependent upon a family of one parameter are obtained. Furthermore, we choose a unique entropy solution through the process of a viscosity vanishing, and obtain a stability for delta-shocks in multidimensions.

**[A]**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****[AH]**R. K. Agarwal and D. W. Halt,*A modified CUSP scheme in wave/particle split form for unstructured grid Euler flows, Frontiers of Computational Fluid Dynamics*, edited by D. A. Caughey and M. M. Hafes, John Wiley and Sons, 1994**[B]**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****[BG]**Yann Brenier and Emmanuel Grenier,*Sticky particles and scalar conservation laws*, SIAM J. Numer. Anal.**35**(1998), no. 6, 2317–2328. MR**1655848**, https://doi.org/10.1137/S0036142997317353**[D]**Constantine M. Dafermos,*Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method*, Arch. Rational Mech. Anal.**52**(1973), 1–9. MR**0340837**, https://doi.org/10.1007/BF00249087**[ERS]**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****[J]**K. T. Joseph,*A Riemann problem whose viscosity solutions contain 𝛿-measures*, Asymptotic Anal.**7**(1993), no. 2, 105–120. MR**1225441****[Ko]**D. J. Korchinski,*Solutions of a Riemann problem for a system of conservation laws possessing classical solutions*, Adelphi University Thesis, 1977**[KK]**Barbara Lee Keyfitz and Herbert C. Kranzer,*A viscosity approximation to a system of conservation laws with no classical Riemann solution*, Nonlinear hyperbolic problems (Bordeaux, 1988) Lecture Notes in Math., vol. 1402, Springer, Berlin, 1989, pp. 185–197. MR**1033283**, https://doi.org/10.1007/BFb0083875**[La]**Peter D. Lax,*Hyperbolic systems of conservation laws and the mathematical theory of shock waves*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR**0350216****[Le]**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**, https://doi.org/10.1007/978-1-4613-9049-7_10**[LC]**Yin Fan Li and Yi Ming Cao,*“Large-particle” difference method with second-order accuracy in gasdynamics*, Sci. Sinica Ser. A**28**(1985), no. 10, 1024–1035. MR**866458****[LL]**J. Li and W. Li,*The Riemann problem for the zero-pressure flow in gas dynamics*, Progress in Natural Sciences, to appear**[LZ]**Jiequan Li and Tong Zhang,*Generalized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations*, Advances in nonlinear partial differential equations and related areas (Beijing, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 219–232. MR**1690831****[M-Z]**Andrew J. Majda, George Majda, and Yu Xi Zheng,*Concentrations in the one-dimensional Vlasov-Poisson equations. I. Temporal development and non-unique weak solutions in the single component case*, Phys. D**74**(1994), no. 3-4, 268–300. MR**1286201**, https://doi.org/10.1016/0167-2789(94)90198-8**[SZe]**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**, https://doi.org/10.1103/RevModPhys.61.185**[SZh]**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**, https://doi.org/10.1090/memo/0654**[TZ]**De Chun Tan and Tong Zhang,*Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws. I. Four-𝐽 cases*, J. Differential Equations**111**(1994), no. 2, 203–254. MR**1284413**, https://doi.org/10.1006/jdeq.1994.1081**[TZZ]**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**, https://doi.org/10.1006/jdeq.1994.1093**[ZZ]**Tong Zhang and Yu Xi Zheng,*Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems*, SIAM J. Math. Anal.**21**(1990), no. 3, 593–630. MR**1046791**, https://doi.org/10.1137/0521032

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
76N99,
35B35,
35L65,
35M20,
35Q35

Retrieve articles in all journals with MSC: 76N99, 35B35, 35L65, 35M20, 35Q35

Additional Information

DOI:
https://doi.org/10.1090/qam/1827367

Article copyright:
© Copyright 2001
American Mathematical Society