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)

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 $H^s({\mathfrak{R}}^{+})$ when the auxiliary data $(\varphi ,h)$ is drawn from $H^s({\mathfrak{R}}^{+})\times H_{\mathrm{loc}}^{\frac{s+1}{3}}({\mathfrak{R}}^{+})$, provided only that $s>-1$ and $s\ne 3m+\frac{1}{2}$ $(m=0,1,2,\dots )$. A similar result is established for (0.1) in $H_{\nu }^s({\mathfrak{R}}^{+})$ provided $(\varphi ,h)$ lies in the space $H_{\nu }^s({\mathfrak{R}}^{+})\times H_{\mathrm{loc}}^{\frac{s+1}{3}}({\mathfrak{R}}^{+})$. Here, $H_{\nu }^s({\mathfrak{R}}^{+})$ 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 $h\in H_{\mathrm{loc}}^{\infty }$ and that for some $\nu >0$ and $s>-1$ with $s\ne 3m+\frac{1}{2}$ $(m=0,1,2,\dots )$, $\varphi$ lies in $H_{\nu }^s({\mathfrak{R}}^{+})$ (respectively $H^s({\mathfrak{R}}^{+})$). Then the corresponding solution $u$ of the IBVP (0.1) (respectively the IBVP (0.2)) belongs to the space $C(0,\infty ;H_{\nu }^{\infty }({\mathfrak{R}}^{+}))$ (respectively $C(0,\infty ;H^{\infty }({\mathfrak{R}}^{+}))$). In particular, for any $s>-1$ with $s\ne 3m+\frac{1}{2}$ $(m=0,1,2,\dots )$, if $\varphi \in H^s({\mathfrak{R}}^{+})$ has compact support and $h\in H_{\mathrm{loc}}^{\infty }({\mathfrak{R}}^{+})$, then the IBVP (0.1) has a unique solution lying in the space $C(0,\infty ;H^{\infty }({\mathfrak{R}}^{+}))$.