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) ut + ux + uux + uxxx = 0, for x, t ≥ 0, u(x, 0) = φ(x), u(0, t) = h(t)  (0.1) for the Korteweg-de Vries (KdV) equation and ut + ux + uux − uxx + uxxx = 0, for x, t ≥ 0, u(x, 0) = φ(x), u(0, t) = h(t)  (0.2) for the Korteweg-de Vries-Burgers (KdV-B) equation. These type of problems arise in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into near-shore zones (see [2] and [5] 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(<+) when the auxiliary data (φ, h) is drawn from Hs(<+) × H s+1 3 loc (<+), provided only that s > −1 with s 6= 3m + 12 (m = 1, 2, · · ·). A similar result is established for (0.1) in H s ν(<) provided (φ, h) lies in the space Hs ν(<) ×H s+1 3 loc (<+). Here, Hs ν(<) is the weighted Sobolev space H ν(<) = {f ∈ H(<); ef ∈ H(<)} with the obvious norm (cf. Kato [40]). 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 6= 3m + 1 2 (m = 1, 2, · · ·), φ lies in Hs ν(<) (respectively Hs(<+)). Then the corresponding solution u of the IBVP (0.1) (respectively the IBVP (0.2)) belongs to the space C(0,∞;H∞ ν (<+)) (respectively C(0,∞;H∞(<+))). In particular, for any s > −1 with s 6= 3m + 12 (m = 1, 2, · · ·), if φ ∈ Hs(<+) has compact support and h ∈ H∞ loc(<), then the IBVP (0.1) has a unique solution lying in the space C(0,∞;H∞(<+)). Acknowledgments. JLB and SMS were partially supported by the National Science Foundation. BYZ was partially supported by the Charles Phelps Taft Memorial Fund at the University of Cincinnati.