A Non-homogeneous Boundary-value Problem for the Korteweg-de Vries Equation in a Quarter Plane

The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem (0.1)  ηt + ηx + ηηx + ηxxx = 0, for x, t ≥ 0, η(x, 0) = φ(x), η(0, t) = h(t), studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local wellposedness is obtained for initial data φ in the class Hs(R+) for s > 3 4 and boundary data h in H (1+s)/3 loc (R +), whereas global well-posedness is shown to hold for φ ∈ Hs(R+), h ∈ H 7+3s 12 loc (R +) when 1 ≤ s ≤ 3, and for φ ∈ Hs(R+), h ∈ H loc (R+) when s ≥ 3. In addition, it is shown that the correspondence that associates to initial data φ and boundary data h the unique solution u of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems. Received by the editors April 19, 2000 and, in revised form, January 8, 2001. 2000 Mathematics Subject Classification. Primary 35Q53; Secondary 76B03, 76B15.