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 \[ (0.1)\qquad \qquad \quad \left \{ \begin {array}{l} \eta _t+\eta _x+\eta \eta _x +\eta _{xxx} =0 , \quad \mbox {for} \ x, t \geq 0, \cr \ \cr \eta (x,0) = \phi (x),\qquad \quad \eta (0,t) =h(t),\end {array}\right . \qquad \qquad \qquad \quad \] 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 $\phi$ in the class $H^s(R^+)$ for $s>\frac 34$ and boundary data $h$ in $H^{(1+s)/3}_{loc} (R^+)$, whereas global well-posedness is shown to hold for $\phi \in H^s (R^+) , \ h\in H^{\frac {7+3s}{12}}_{loc} (R^+)$ when $1\leq s\leq 3$, and for $\phi \in H^s(R^+) , \ h\in H^{(s+1)/3}_{loc} (R^+)$ when $s\geq 3$. In addition, it is shown that the correspondence that associates to initial data $\phi$ 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.