Non-homogeneous boundary value problems for the Korteweg–de Vries and the Korteweg–de Vries–Burgers equations in a quarter plane
Authors: Sun S.M. , Zhang Bing-Yu , Bona Jerry L. 
Published: January 2008

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}^{ + }))

.