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)
Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics. (English) Zbl 1153.76051
Summary: The conservation laws for second-order scalar partial differential equations and systems of partial differential equations which occur in fluid mechanics are constructed using different approaches. The direct method, Noether’s theorem, the characteristic method, the variational approach (multiplier approach) for arbitrary functions as well as on the solution space, symmetry conditions on the conserved quantities, the direct construction formula approach, the partial Noether approach and the Noether approach for the equation and its adjoint are discussed and explained with the help of an illustrative example on a nonlinear field equation describing the relaxation to a Maxwellian distribution. The conservation laws for the nonlinear diffusion equation for the spreading of an axisymmetric thin liquid drop, the system of two partial differential equations governing flow in a laminar two-dimensional jet and the system of two partial differential equations governing flow in a laminar radial jet are discussed via these approaches.

MSC:
76M60Symmetry analysis, Lie group and algebra methods (fluid mechanics)
76R50Diffusion (fluid mechanics)
WorldCat.org
Full Text: DOI
References:
[1] R. Naz, D.P. Mason, F.M. Mahomed, Conservation laws and physical conserved quantities for laminar two-dimensional and radial jets, J. Nonlinear Anal. B: Real World Appl., submitted for publication. · Zbl 1177.35171
[2] Noether, E.: Invariante variationsprobleme. Nacr. konig. Gesell. wissen., gottingen, math.-phys. Kl. heft 2, 235-257 (1918)
[3] P.S. Laplace (English translation, Celestrial Mechanics, New York, 1966), 1798.
[4] Steudel, H.: Uber die zuordnung zwischen invarianzeigenschaften und erhaltungssatzen. Z. naturforsch. 17A, 129-132 (1962)
[5] Olver, P. J.: Applications of Lie groups to differential equations. (1993) · Zbl 0785.58003
[6] Anco, S. C.; Bluman, G. W.: Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications. Eur. J. Appl. math. 13, 545-566 (2002) · Zbl 1034.35070
[7] Anco, S. C.; Bluman, G. W.: Direct construction method for conservation laws of partial differential equations. Part II: General treatment. Eur. J. Appl. math. 9, 567-585 (2002) · Zbl 1034.35071
[8] Wolf, T.: A comparison of four approaches to the calculation of conservation laws. Eur. J. Appl. math. 13, 129-152 (2002) · Zbl 1002.35008
[9] Kara, A. H.; Mahomed, F. M.: Relationship between symmetries and conservation laws. Int. J. Theor. phys. 39, 23-40 (2000) · Zbl 0962.35009
[10] Kara, A. H.; Mahomed, F. M.: Noether-type symmetries and conservation laws via partial lagragians. Nonlinear dynam. 45, 367-383 (2006) · Zbl 1121.70014
[11] Atherton, R. W.; Homsy, G. M.: On the existence and formulation of variational principles for nonlinear differential equations. Studies appl. Math. 54, 31-60 (1975) · Zbl 0322.49019
[12] Ibragimov, N. H.: A new conservation theorem. J. math. Anal. appl. 333, 311-328 (2007) · Zbl 1160.35008
[13] Euler, N.; Leach, P. G.; Mahomed, F. M.; Steeb, W. -H.: Symmetry vector fields and similarity solutions of nonlinear field equation describing the relaxation to a Maxwell distribution. Int. J. Theor. phys. 27, 717-723 (1988) · Zbl 0697.35141
[14] Momoniat, E.; Mason, D. P.; Mahomed, F. M.: Non-linear diffusion of an axisymmetric thin liquid drop: group invariant solution and conservation law. Int. J. Nonlinear mech. 36, 879-885 (2001) · Zbl 05137964
[15] Schlichting, H.: Boundary layer theory. (1968) · Zbl 0096.20105
[16] R. Naz, D.P. Mason, F.M. Mahomed, Conservation laws via the partial Lagrangian and group invariant solutions for radial and two-dimensional free jets, J. Nonlinear Anal. B: Real World Appl., submitted for publication. · Zbl 1269.76022
[17] Schwarz, W. H.: The radial free jet. Chem. eng. Sci. 18, 779-786 (1963)