An existence theory for water waves and the Boussinesq and Korteweg- deVries scaling limits. (English) Zbl 0577.76030

Craig, Walter

Commun. Partial Differ. Equations 10, 787-1003 (1985).

This comprehensive work aims at clarifying the correspondence between solutions of the problem of waves in an inviscid and incompressible fluid (water) and those of the system described by the Boussinesq equation and the Korteweg-deVries equation. A theorem of local existence in time and of uniqueness for solutions of the water wave problem, given initial data in some Sobolev space, is established in the long-wave regime. The long- time existence result constitutes an important ingredient in describing a rigorous correspondence of the water wave solutions in the two long-wave - i.e. Boussinesq and KdV - scaling limits. The long-wave approximation is justified in this sense of modeling water waves.
The Hilbert transform for the variable fluid region, defined on the free surface of the fluid, is a singular integral operator being of central importance for the methods used in this paper. An analysis of this operator, including a uniform expansion of this operator in terms of the surface deformations and their difference quotients, is presented within the framework of this paper.

Reviewer: M.Biermann

MSC:

76B25

Solitary waves for incompressible inviscid fluids

Keywords:

scaling limit; Boussinesq equation; Korteweg-deVries equation; existence; uniqueness; Sobolev space; long-wave approximation; Hilbert transform; singular integral operator

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