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• W1866709725 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 $phi in H^s (R^+) , hin H^{frac {7+3s}{12}}_{loc} (R^+)$ when $1leq sleq 3$, and for $phi in H^s(R^+) , hin H^{(s+1)/3}_{loc} (R^+)$ when $sgeq 3$. 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." @default.
• W1866709725 created "2016-06-24" @default.
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• W1866709725 date "2001-09-26" @default.
• W1866709725 modified "2023-10-14" @default.
• W1866709725 title "A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane" @default.
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• W1866709725 doi "https://doi.org/10.1090/s0002-9947-01-02885-9" @default.
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