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-boundary-value problems (IBVPs) ut+ux+uux+uxxx=0, for x, t ≥ 0,} u(x, 0) = Φ(x), u(0,t)=h(t) for the Korteweg-de Vries (KdV) equation and u1 + ux + uux - uxx + uxxxx = 0, for x,t ≥0,} (0.2) u(x, 0)= Φ(x), u(0, t) = h(t) 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 (1981) 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 Hs(R+) when the auxiliary data (Φ, h) is drawn from Hs(R+) x HI=1 3 loc (R+), provided only that s > -1 and s ≠ 3m + 1/2 (m = 0, 1, 2,...). A similar result is established for (0.1) in Hsν(R+) provided (Φ, h) lies in the space Hsν(Ρ+) x Hs+13loc (R+). Here, Hsν(R+) is the weighted Sobolev space Hsν(R+)= {f ∈ Hs (R+); eνx f ∈ Hs (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 ∈ H∞loc and that for some v > 0 and s > -1 with s ≠ 3m + 1/2 (m = 0, 1, 2,...), Φ lies in Hsν(R+) (respectively Hs(R+)). Then the corresponding solution u of the IBVP (0.1) (respectively the IBVP (0.2)) belongs to the space C(0, ∞; H∞ν(R+)) (respectively C(0, ∞; H∞(R+))). In particular, for any s > -1 with s ≠ 3m + (m = 0, 1, 2,...). if Φ ∈ Hs (R+) has compact support and h e H∞loc (R+ ), then the IBVP (0.1) has a unique solution lying in the space C(0, ∞: H∞(R+)).