zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model. (English) Zbl 1211.35193
The authors consider a 2×2 system of conservation laws, which is an perturbation of the pressureless gas dynamics system. Particularly, they study the limiting behavior of the solutions of the Riemann problems. It is proved that for the system of the perturbed Aw-Rascle model, the limits of its Riemann solution are the Riemann solution of the pressureless gas dynamics with same initial data. In the meantime, the formation of delta-shocks and vacuum are also discussed.
MSC:
35L65Conservation laws
35L67Shocks and singularities
35B25Singular perturbations (PDE)
90B20Traffic problems
References:
[1]Agarwal, R. K.; Halt, D. W.: A modified CUSP scheme in wave/particle split form for unstructured grid Euler flows, Frontiers of computational fluid dynamics, 155-163 (1994)
[2]Aw, A.; Klar, A.; Materne, A.; Rascle, M.: Derivation of continuum traffic flow models from microscopic follow-the-leader model, SIAM J. Appl. math. 63, 259-278 (2002) · Zbl 1023.35063 · doi:10.1137/S0036139900380955
[3]Aw, A.; Rascle, M.: Resurrection of second order models of traffic flow, SIAM J. Appl. math. 60, 916-938 (2000) · Zbl 0957.35086 · doi:10.1137/S0036139997332099
[4]Berthelin, F.; Degond, P.; Delitata, M.; Rascle, M.: A model for the formation and evolution of traffic jams, Arch. ration. Mech. anal. 187, 185-220 (2008) · Zbl 1153.90003 · doi:10.1007/s00205-007-0061-9
[5]Berthelin, F.; Degond, P.; Leblanc, V.; Moutari, S.; Rascle, M.; Royer, J.: A traffic-flow model with constraints for the modeling of traffic jams, Math. models methods appl. Sci. 18, 1269-1298 (2008) · Zbl 1197.35159 · doi:10.1142/S0218202508003030
[6]Bouchut, F.: On zero pressure gas dynamics, Ser. adv. Math. appl. Sci. 22, 171-190 (1994) · Zbl 0863.76068
[7]Brenier, Y.; Grenier, E.: Sticky particles and scalar conservation laws, SIAM J. Numer. anal. 35, 2317-2328 (1998) · Zbl 0924.35080 · doi:10.1137/S0036142997317353
[8]Chang, T.; Chen, G. Q.; Yang, S.: On the Riemann problem for two-dimensional Euler equations. I. interaction of shocks and rarefaction waves, Discrete contin. Dyn. syst. 1, 555-584 (1995) · Zbl 0874.76031 · doi:10.3934/dcds.1995.1.555
[9]Chang, T.; Chen, G. Q.; Yang, S.: On the Riemann problem for two-dimensional Euler equations. II. interaction of contact discontinuities, Discrete contin. Dyn. syst. 6, 419-430 (2000) · Zbl 1032.76062 · doi:10.3934/dcds.2000.6.419
[10]Chang, T.; Hsiao, L.: The Riemann problem and interaction of waves in gas dynamics, Pitman monogr. Surveys pure appl. Math. 41 (1989) · Zbl 0698.76078
[11]Chen, G. Q.; Liu, H.: Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. anal. 34, 925-938 (2003) · Zbl 1038.35035 · doi:10.1137/S0036141001399350
[12]Chen, G. Q.; Liu, H.: Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Phys. D 189, 141-165 (2004) · Zbl 1098.76603 · doi:10.1016/j.physd.2003.09.039
[13]Dafermos, C. M.: Hyperbolic conservation laws in continuum physics, Grundlehren math. Wiss. (2000)
[14]Daganzo, C.: Requiem for second order fluid approximations of traffic flow, Transportation res. Part B 29, 277-286 (1995)
[15]Dal Maso, G.; Lefloch, P. G.; Murat, F.: Definition and weak stability of nonconservative products, J. math. Pures appl. 74, 483-548 (1995) · Zbl 0853.35068
[16]Danilov, V. G.; Shelkovich, V. M.: Dynamics of propagation and interaction of δ-shock waves in conservation law systems, J. differential equations 221, 333-381 (2005) · Zbl 1072.35121 · doi:10.1016/j.jde.2004.12.011
[17]Danilov, V. G.; Shelkovich, V. M.: Delta-shock waves type solution of hyperbolic systems of conservation laws, Quart. appl. Math. 63, 401-427 (2005)
[18]E., W.; Rykov, Yu.G.; Sinai, Ya.G.: 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, 349-380 (1996) · Zbl 0852.35097 · doi:10.1007/BF02101897
[19]Garavello, M.; Piccoli, B.: Traffic flow on a road network using the aw-rascle model, Comm. partial differential equations 31, 243-275 (2006) · Zbl 1090.90032 · doi:10.1080/03605300500358053
[20]Greenberg, J. M.: Extensions and amplifications on a traffic model of aw and rascle, SIAM J. Appl. math. 62, 729-745 (2001) · Zbl 1006.35064 · doi:10.1137/S0036139900378657
[21]Greenberg, J. M.; Klar, A.; Rascle, M.: Congestion on multilane highways, SIAM J. Appl. math. 63, 818-833 (2003) · Zbl 1024.35064 · doi:10.1137/S0036139901396309
[22]Hayes, B. T.; Lefloch, P. G.: Measure solutions to a strictly hyperbolic system of conservation laws, Nonlinearity 9, 1547-1563 (1996) · Zbl 0908.35075 · doi:10.1088/0951-7715/9/6/009
[23]Helbing, D.; Johansson, A. F.: On the controversy around daganzo’s requiem for the aw-rascle’s resurrection of second-order traffic flow models, Eur. phys. J. B 69, 549-562 (2009)
[24]Herty, M.; Rascle, M.: Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. anal. 38, 595-616 (2006)
[25]Huang, F.: Weak solution to pressureless type system, Comm. partial differential equations 30, 283-304 (2005) · Zbl 1074.35021 · doi:10.1081/PDE-200050026
[26]Huang, F.; Wang, Z.: Well-posedness for pressureless flow, Comm. math. Phys. 222, 117-146 (2001) · Zbl 0988.35112 · doi:10.1007/s002200100506
[27]Keyfitz, B. L.; Kranzer, H. C.: Spaces of weighted measures for conservation laws with singular shock solutions, J. differential equations 118, 420-451 (1995) · Zbl 0821.35096 · doi:10.1006/jdeq.1995.1080
[28]D.J. Korchinski, Solution of a Riemann problem for a system of conservation laws possessing no classical weak solution, Thesis, Adelphi University, 1977.
[29]Lefloch, P. G.; Liu, T. P.: Existence theory to nonlinear hyperbolic systems under nonconservative form, Forum math. 5, 261-280 (1993) · Zbl 0804.35086 · doi:10.1515/form.1993.5.261
[30]Li, J.: Note on the compressible Euler equations with zero temperature, Appl. math. Lett. 14, 519-523 (2001) · Zbl 0986.76079 · doi:10.1016/S0893-9659(00)00187-7
[31]Li, J.; Zhang, T.; Yang, S.: The two-dimensional Riemann problem in gas dynamics, Pitman monogr. Surveys pure appl. Math. 98 (1998) · Zbl 0935.76002
[32]Li, Y.; Cao, Y.: Second order large particle difference method, Sci. China ser. A 8, 1024-1035 (1985)
[33]Lions, P. L.; Perthame, B.; Tadmor, E.: Kinetic formulation of the isentropic gas dynamics and p-system, Comm. math. Phys. 163, 415-431 (1994) · Zbl 0799.35151 · doi:10.1007/BF02102014
[34]Liu, T. P.; Smoller, J.: On the vacuum state for isentropic gas dynamic equations, Adv. in appl. Math. 1, 345-359 (1980) · Zbl 0461.76055 · doi:10.1016/0196-8858(80)90016-0
[35]Mitrovic, D.; Nedeljkov, M.: Delta-shock waves as a limit of shock waves, J. hyperbolic differ. Equ. 4, 629-653 (2007) · Zbl 1145.35086 · doi:10.1142/S021989160700129X
[36]Moutari, S.; Rascle, M.: A hybrid Lagrangian model based on the aw-rascle traffic flow model, SIAM J. Appl. math. 68, 413-436 (2007) · Zbl 1148.35049 · doi:10.1137/060678415
[37]Panov, E. Yu.; Shelkovich, V. M.: δ’-shock waves as a new type of solutions to system of conservation laws, J. differential equations 228, 49-86 (2006) · Zbl 1108.35116 · doi:10.1016/j.jde.2006.04.004
[38]Serre, D.: Systems of conservation laws 1/2, (1999)
[39]Sever, M.: A class of nonlinear, nonhyperbolic systems of conservation laws with well-posed initial value problem, J. differential equations 180, 238-271 (2002) · Zbl 1012.35056 · doi:10.1006/jdeq.2001.4060
[40]Shandarin, S. F.; Zeldovich, Ya.B.: The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium, Rev. modern phys. 61, 185-220 (1989)
[41]Shelkovich, V. M.: δ- and δ’-shock wave types of singular solutions of systems of conservation laws and transport and concentration processes, Russian math. Surveys 63, 473-546 (2008) · Zbl 1194.35005 · doi:10.1070/RM2008v063n03ABEH004534
[42]Sheng, W.; Zhang, T.: The Riemann problem for the transportation equations in gas dynamics, Mem. amer. Math. soc. 137(654) (1999)
[43]Smoller, J.: Shock waves and reaction-diffusion equations, (1994)
[44]Sun, M.: Interactions of elementary waves for the aw-rascle model, SIAM J. Appl. math. 69, 1542-1558 (2009) · Zbl 1184.35208 · doi:10.1137/080731402
[45]Tan, D.; Zhang, T.; Zheng, Y.: Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. differential equations 112, 1-32 (1994) · Zbl 0804.35077 · doi:10.1006/jdeq.1994.1093
[46]Temple, B.: Systems of conservation laws with invariant submanifolds, Trans. amer. Math. soc. 280, 781-795 (1983) · Zbl 0559.35046 · doi:10.2307/1999646
[47]Yang, H.: Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. differential equations 159, 447-484 (1999) · Zbl 0948.35079 · doi:10.1006/jdeq.1999.3629
[48]Zeldovich, Y. B.; Myshkis, A. D.: Elements of mathematical physics: medium consisting of non-interacting particles, (1973)
[49]Zhang, H. M.: A non-equilibrium traffic model devoid of gas-like behavior, Transportation res. Part B 36, 275-290 (2002)