A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane

Abstract

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) {η t + η x + ηη x + η xxx = 0, for x, t ≥ 0. (0.1) {η t + η x + ηη x + η xxx = 0, for x, t ≥ 0 η(x,0) = o(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 well-posedness is obtained for initial data o in the class H s (R + ) for s > 3 and boundary data h in H (1+s)/3 loc (R + ), whereas global well-posedness is shown to hold for o ∈ H s (R + ), h ∈ H 12 loc (R + ) when 1 3. In addition, it is shown that the correspondence that associates to initial data O 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.