zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Interactions of delta shock waves in a strictly hyperbolic system of conservation laws. (English) Zbl 1155.35059
The authors consider the Riemann problem for a strictly hyperbolic system of two conservation laws from magnetohydrodynamics. The system is considered from the point of view of weak solutions. Generally a solution to such problem is a combination of shock waves, contact discontinuities, rarefaction waves, and delta shock waves. It is shown how the solutions continue beyond points of interaction of these types of solutions. A type of generalized solution is introduced: delta contact discontinuity. A complete description of all global solutions with initial data given by two jump points and three constant states is presented.

35L65Conservation laws
35L67Shocks and singularities
[1]Bressan, A.: Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem, Oxford lecture ser. Math. appl. 20 (2000) · Zbl 0997.35002
[2]Dafermos, M.: Hyperbolic conservation laws in continuum physics, Grundlehren math. Wiss. 325 (2005)
[3]Hayes, B. T.; Le Floch, 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
[4]Huang, F.: Weak solutions to pressureless type systems, Comm. partial differential equations 30, 283-304 (2005) · Zbl 1074.35021 · doi:10.1081/PDE-200050026
[5]Hurd, A. E.; Sattinger, D. H.: Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients, Trans. amer. Math. soc. 132, 159-174 (1968) · Zbl 0155.16401 · doi:10.2307/1994888
[6]Keyfitz, B. L.: Conservation laws, delta shocks and singular shocks, Chapman & Hall/CRC res. Notes math. 401, 99-111 (1999) · Zbl 0933.35134
[7]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
[8]Keyfitz, B. L.; Sever, M.; Zhang, F.: Viscous singular shock structure for a nonhyperbolic two-fluid model, Nonlinearity 17, 1731-1747 (2004) · Zbl 1077.35091 · doi:10.1088/0951-7715/17/5/010
[9]D.J. Korchinski, Solution of a Riemann problem for a 2nbsp;2 system of conservation laws possessing no classical weak solution, PhD thesis, Adelphi University, Garden City, New York, 1977
[10]Lax, P.: Hyperbolic systems of conservation laws II, Comm. pure appl. Math. 10, 537-566 (1957)
[11]Mitrović, D.; Nedeljkov, M.: Delta shock waves as a limit of shock waves, J. hyperbolic differ. Equ. 4, 1-25 (2007) · Zbl 1145.35086 · doi:10.1142/S021989160700129X
[12]Nedeljkov, M.: Delta and singular delta locus for one dimensional systems of conservation laws, Math. methods appl. Sci. 27, 931-955 (2004) · Zbl 1056.35115 · doi:10.1002/mma.480
[13]Oberguggenberger, M.: Multiplication of distributions and applications to partial differential equations, Pitman res. Notes math. Ser. 259 (1992) · Zbl 0818.46036
[14]Oberguggenberger, M.: Case study of a nonlinear, nonconservative, nonstrictly hyperbolic system, Nonlinear anal. 19, 53-79 (1992) · Zbl 0790.35065 · doi:10.1016/0362-546X(92)90030-I
[15]Sever, M.: Viscous structure of singular shocks, Nonlinearity 15, 705-725 (2002) · Zbl 1026.35080 · doi:10.1088/0951-7715/15/3/311
[16]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
[17]Whitham, G. B.: Linear and nonlinear waves, (1974) · Zbl 0373.76001