Non-homogeneous boundary value problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane

Attention is given to the initial-bounclary-value problems (IBVPs) [GRAPHICS] for the Korteweg-de Vries (KdV) equation and [GRAPHICS] for the Korteweg-de Vries-Burgers (KdV-B) equation. These types of problems arise in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into near-shore zones (see [B. Boczar-Karakiewicz, J.L. Bona, Wave dominated shelves: a model of sand ridge formation by progressive infragravity waves, in: R.J. Knight, J.R. McLean (Eds.), Shelf Sands and Sandstones, in: Canadian Society of Petroleum Geologists Memoir, vol. 11, 1986, pp. 163-179] and [J.L. Bona, W.G. Pritchard, L.R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Roy. Soc. London Ser. A 302 (198 1) 457-510] for example). Our concern here is with the mathematical theory appertaining to these problems. Improving upon the existing results for (0.2). we show this problem to be (locally) well-posed in H-s(R+) when the auxiliary data (phi, h) is drawn from H-s ((R)(+) x H-loc(s+1/3)(R+), provided only that s > -1 and s not equal 3m + 1/2 (m = 0.1.2...). A similar result is established for (0.1) in H-v(s)(R+) provided (phi . h) lies in the space H-v(s)(R+) x H-loc(s+1/3)(R+). Here H-v(s)(R+) is the weighted Sovolev space H-v(s)(R+) = {f is an element of H-s (R+); e(vx) f is an element of H-s (R+)} with the obvious norm (cf. Kato [T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equations, in: Advances in Mathematics Supplementary Studies, in: Studies Appl. Math., vol. 8, 1983, pp. 93-128]). Both local and global in time results are derived. An added outcome of our analysis is a very strong smoothing property associated with the problems < 0.1 > and (0.2) which may be expressed as follows. Suppose h is an element of H-loc(infinity) and that for somve v > 0 and s > -1 with s not equal 30 + 1/2 (m = 0.1.2....). phi lies in H-v(s) (R+)), Then the corresponding solution it of the IBVP (0. 1) (respectively the IBVP (0.2)) belongs to the space C(0,infinity: H-infinity (R+)). (respectively C(0,infinity: H-infinity (R+))). In particular, for any s > -1 with s not equal 3m + 1/2 (m = 0, 1, 2,...), if phi is an element of H-s(R+) has compact support and h is an element of H-loc(infinity)(R+), then the IBVP (0.1) has a unique solution lying in the space C(0,infinity: H-infinity (R+)) (C) 2008 Elsevier Masson SAS. All rights reserved.