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

Abstract

Attention is given to the initial-boundary-value problems (IBVPs)(0.1)ut+ux+uux+uxxx=0,forx,t⩾0,u(x,0)=ϕ(x),u(0,t)=h(t)} for the Korteweg–de Vries (KdV) equation and(0.2)ut+ux+uux−uxx+uxxx=0,forx,t⩾0,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+)×Hlocs+13(R+), provided only that s>−1 and s≠3m+12(m=0,1,2,…). A similar result is established for (0.1) in Hνs(R+) provided (ϕ,h) lies in the space Hνs(R+)×Hlocs+13(R+). Here, Hνs(R+) is the weighted Sobolev spaceHνs(R+)={f∈Hs(R+);eνxf∈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∈Hloc∞ and that for some ν>0 and s>−1 with s≠3m+12(m=0,1,2,…), ϕ lies in Hνs(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,∞;Hν∞(R+)) (respectively C(0,∞;H∞(R+))). In particular, for any s>−1 with s≠3m+12(m=0,1,2,…), if ϕ∈Hs(R+) has compact support and h∈Hloc∞(R+), then the IBVP (0.1) has a unique solution lying in the space C(0,∞;H∞(R+)).