Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0897.35067
Toland, J.F.
Stokes waves. (English)
[J] Topol. Methods Nonlinear Anal. 7, No.1, 1-48 (1996); errata ibid. 8, No.2, 413-414 (1996). ISSN 1230-3429

In this consistent paper, the author gives complete proofs of the main results about the Stokes waves problem starting with its basic formulation as a free boundary problem for a harmonic function in an unknown domain in the plane.\par With the assumptions that the boundary is a real analytic curve and that the complex potential must have an analytic extension across the boundary, the equations can be manipulated at will and various estimates on the wave slope and speed emerge from calculation involving the maximum principle for harmonic functions in the plane.\par To formulate the boundary value problem in a way which is amenable to existence theory the author uses the method of hodograph transformation to map the unknown domain occupied by the water into a fixed semi-infinite strip in a complex plane where the variable is the complex potential. So the problem is formulated as Nekrasov's integral equation in a cone in a Banach space of continuous functions. A global existence theory for Nekrasov's integral equation is obtained from classical global bifurcation theory. Then the behaviour of large amplitude Stokes waves is examined.\par Finally the paper gives a very brief description of how the present understanding fails to predict numerically observed secondary and subsequent bifurcation.
[V.A.Sava (Iaşi)]

MSC 2000:: *35Q35 Other equations arising in fluid mechanics
76B15 Wave motions (fluid mechanics)
45G10 Nonsingular nonlinear integral equations
31A10 Integral representations of harmonic functions (two-dimensional)
47J05 Equations involving nonlinear operators (general)

Keywords: Stokes waves; harmonic functions; existence theory; hodograph transformation; complex potential; bifurcation; Nekrasov's integral equation

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]