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

### S.M. Sun

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-4097, USA### Bing-Yu Zhang

Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA### Jerry L. Bona

Department of Mathematics, Statistics & Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA

## Abstract

Attention is given to the initial-boundary-value problems (IBVPs)

for the Korteweg–de Vries (KdV) equation and

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 $H_{s}(R_{+})$ when the auxiliary data $(ϕ,h)$ is drawn from $H_{s}(R_{+})×H_{loc}(R_{+})$, provided only that $s>−1$ and $s=3m+21 $ $(m=0,1,2,…)$. A similar result is established for (0.1) in $H_{ν}(R_{+})$ provided $(ϕ,h)$ lies in the space $H_{ν}(R_{+})×H_{loc}(R_{+})$. Here, $H_{ν}(R_{+})$ is the weighted Sobolev space

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 $ν>0$ and $s>−1$ with $s=3m+21 $ $(m=0,1,2,…)$, $ϕ$ lies in $H_{ν}(R_{+})$ (respectively $H_{s}(R_{+})$). 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+21 $ $(m=0,1,2,…)$, if $ϕ∈H_{s}(R_{+})$ has compact support and $h∈H_{loc}(R_{+})$, then the IBVP (0.1) has a unique solution lying in the space $C(0,∞;H_{∞}(R_{+}))$.

## Cite this article

S.M. Sun, Bing-Yu Zhang, Jerry L. Bona, Non-homogeneous boundary value problems for the Korteweg–de Vries and the Korteweg–de Vries–Burgers equations in a quarter plane. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 6, pp. 1145–1185

DOI 10.1016/J.ANIHPC.2007.07.006