×

zbMATH — the first resource for mathematics

Vorticity and the mathematical theory of incompressible fluid flow. (English) Zbl 0595.76021
The problems associated with vortex dynamics and high Reynolds number incompressible fluid flow are certainly of intense scientific interest with numerous applications ranging from accurate prediction of hurricane paths to efficient design of internal combustion engines to control of the hazardous large vortices shed by landing jumbo jets - the mechanisms responsible for the phenomena in these problems remain poorly understood.
The paper reports on recent and ongoing work in the theory of vortex dynamics in high Reynolds number and inviscid incompressible fluid flow with an emphasis on the interaction of ideas from numerical, asymptotic, and qualitative modelling as well as rigorous proofs for prototype problems. Besides being a report, a new consistent point of view on the topic will be presented, several results from ongoing research will be stated, and a few new proofs will be given.

MSC:
76B47 Vortex flows for incompressible inviscid fluids
76Bxx Incompressible inviscid fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, J. Comp. Phys. 61 pp 417– (1985)
[2] An Introduction to Fluid Mechanics, Cambridge University Press, 1970.
[3] Beale, Math. Comp. 39 pp 29– (1982)
[4] Beale, Comm. Math. Phys. 94 pp 61– (1984)
[5] Brachet, J. Fluid Mech. 130 pp 411– (1983)
[6] and , Long-time existence for a perturbed vortex sheet, preprint, December, 1985.
[7] Chorin, Comm. Pure Appl. Math. 34 pp 853– (1981)
[8] Chorin, Comm. Math. Phys. 83 pp 517– (1982)
[9] and , An Introduction to Mathematical Fluid Mechanics, Springer New York, 1979. · doi:10.1007/978-1-4684-0082-3
[10] Constantin, Comm. Math. Physics 104 pp 311– (1986)
[11] Constantin, Comm. Pure Appl. Math. 38 pp 715– (1985)
[12] DiPerna, Comm. Math. Physics.
[13] Ph. Dissertation, Geophysical Fluid Dynamics Laboratory, Princeton, 1985.
[14] Yudovich, Zh. Vych. Mat. 3 pp 1032– (1963)
[15] Kato, J. Funct. Anal. 9 pp 296– (1972)
[16] Desingularization of periodic vortex sheet roll-up, J. Comp. Phys. (in press). · Zbl 0591.76059
[17] Computation of vortex sheet roll-up in the Treffitz plane, preprint, March 1986.
[18] Hydrodynamics, Dover, New York, 1945.
[19] and , Fluid Mechanics, Addison-Wesley, 1959.
[20] Mathematical Foundations of Incompressible Fluid Flow, Lecture Notes, Princeton Math. Dept., 1985.
[21] Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer, Appl. Math. Sci. #53, 1984. · doi:10.1007/978-1-4612-1116-7
[22] Intermittent turbulence and fractal dimension in Turbulence and Navier-Stokes Equations, (editor), Springer, New York, 1975.
[23] Morf, Phys. Rev. Lett. 44 pp 572– (1980)
[24] Neu, J. Fluid Mech. 143 pp 253– (1984)
[25] Ph. D. Dissertation, 1986, Univ. California, Berkeley and Princeton Univ.
[26] Ponce, Comm. Math. Phys. 98 pp 349– (1985)
[27] Saffman, Ann. Rev. Fluid Mech. 11 pp 95– (1979)
[28] Singular Integrals and the Differentiability Properties of Functions, Princeton Univ. Press, 1970.
[29] Sulem, Comm. Math. Phys. 80 pp 485– (1981)
[30] , and , Contour dynamics for the Euler equations in two dimensions, J. Comp. Phys. 1979, pp. 96–106. · Zbl 0405.76014
[31] Siggia, Phys. Fluids 28 pp 794– (1985)
[32] and , Concentrations in regularizations for 2-D incompressible flow, submitted to Comm. Pure Appl. Math.
[33] and , Reduced Hausdorff dimension and concentration-cancellation for 2-D incompressible flow, preprint, September 1986.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.