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) \begin{matrix} u_{t} + u_{x} + uu_{x} + u_{xxx} = 0,\:\text{for}\:x,t⩾0, \\ u(x,0) = \phi (x),\:u(0,t) = h(t) \\ \end{matrix} for the Korteweg–de Vries (KdV) equation and \begin{matrix} u_{t} + u_{x} + uu_{x}−u_{xx} + u_{xxx} = 0,\:\text{for}\:x,t⩾0, \\ u(x,0) = \phi (x),\:u(0,t) = h(t) \\ \end{matrix} 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 (\phi ,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 \neq 3m + \frac{1}{2} (m = 0,1,2,…) . A similar result is established for (0.1) in H_{\nu }^{s}(\mathfrak{R}^{ + }) provided (\phi ,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 H_{\nu }^{s}\left(\mathfrak{R}^{ + }\right) = \left\{f \in H^{s}\left(\mathfrak{R}^{ + }\right);\:e^{\nu x}f \in H^{s}\left(\mathfrak{R}^{ + }\right)\right\} 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 \neq 3m + \frac{1}{2} (m = 0,1,2,…) , \phi 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 \neq 3m + \frac{1}{2} (m = 0,1,2,…) , if \phi \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}^{ + })) .

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