Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T19:17:09.479Z Has data issue: false hasContentIssue false
  • English
  • Français

A one dimensional analogue of the vorticity equation

Published online by Cambridge University Press:  17 April 2009

K. Sriskandarajah
K. Sriskandarajah
Affiliation:
Department of MathematicsMonash UniversityClayton Vic 3168Australia e-mail: sris@wave.maths.monash.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the qualitative properties of the one dimensional analogue of the Helmholtz vorticity advection equation. The second order hyperbolic equation has the unusual characteristic of disturbances propagating at infinite speed. The global solution for Goursat data is given in closed form. We also obtain qualitative results on the nodal curve where the solution is zero. A related perturbation problem is considered and solutions for small data are obtained. The forced vorticity equation admits a class of soliton solutions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 1 , August 1997 , pp. 119 - 134
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Angennent, S., ‘The zero set of a solution of a parabolic equation’, J. Reine. Angew. Math. 390 (1988), 79–96.Google Scholar
[2]Calogero, F., ‘A solvable nonlinear wave equation’, Stud. Appl. Math. (1984), 189–199.Google Scholar
[3]Constantin, P., Lax, P.D. and Majda, A., ‘A simple one-dimensional model for the three-dimensional vorticity equation’, Comm. Pure Appl. Math. 38 (1985), 715–724.CrossRefGoogle Scholar
[4]Drazin, P.G., Solitons, London Mathematical Society Lecture Notes Series 85 (Cambridge University Press, Cambridge, New York, 1983).CrossRefGoogle Scholar
[5]He, Y. and Moodie, T.B., ‘A transonic model problem’, in Nonlinear diffusion phenomenon, (Sachdev, P.L. and Grundy, R.E., Editors) (Norasa Publishing House, New Delhi, 1992), pp. 92–117.Google Scholar
[6]Liu, T.P., ‘Nonlinear resonance for quasilinear hyperbolic equation’, J. Math. Phys. 28 (1987), 243–260.CrossRefGoogle Scholar
[7]Majda, A., ‘Vorticity and the mathematical theory of incompressible fluid flow’, Comm. Pure Appl. Math. 39 (1986), S186–S220.Google Scholar
[8]Matano, H., ‘Nonincrease of the lap number of a solution for a one dimensional semilinear parabolic equation’, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 29 (1982), 401–411.Google Scholar
[9]Schochet, S., ‘Explicit solutions of the viscous model vorticity equation’, Comm. Pure Appl. Math. 39 (1986), 531–537.CrossRefGoogle Scholar