Abstract. The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem (0.1)studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0

Attention is given to the initial-boundary-value problems (IBVPs) u t + u x + uu x + u xxx = 0, for x, t 0,for the Korteweg-de Vries (KdV) equation andfor 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.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 ( + ) when the auxiliary data (φ, h) is drawn from H s ( + ) × H s+1 3 (m = 0, 1, 2, . . .), φ lies in H s ν ( + ) (respectively H s ( + )). 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 = 3m + 1 2 (m = 0, 1, 2, . . .), if φ ∈ H s ( + ) has compact support and h ∈ H ∞ loc ( + ), then the IBVP (0.1) has a unique solution lying in the space C(0, ∞; H ∞ ( + )).

This paper is concerned with initial-boundary-value problems (IBVPs) for a class of nonlinear Schrödinger equations posed either on a half line R + or on a bounded interval (0, L) with nonhomogeneous boundary conditions. For any s with 0 ≤ s < 5/2 and s = 3/2, it is shown that the relevant IBVPs are locally well-posed if the initial data lie in the L 2based Sobolev spaces H s (R + ) in the case of the half line and in H s (0, L) on a bounded interval, provided the boundary data are selected from H * Corresponding author on R, while the slightly higher regularity of boundary data for the IBVP on (0, L) resembles what is found for temporal traces of spatially periodic solutions.

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Fatigue strength, crack initiation and small crack growth behaviour in two kinds of squeezecast aluminium alloys, AC8A-T6 and AC4C-T6 were investigated using smooth specimens subjected to rotatary-bending fatigue at room temperature. Fatigue resistance of these alloys was almost the same as that of the wrought aluminium alloys because of their fine microstructure and of the decrease in defect size due to squeeze-casting. Fatigue crack initiation sites were at the eutectic silicon particles on the surface of specimens or at internal microporosity in the specimens. Crack initiation life, dehed as a crack length of 50 pm on the specimen surfam, was successfully estimated from an evaluation of initiation sites using fracture mechanics and the statistics of extrema. Small fatigue crack growth in the two kinds of alloys obeys the relation proposed by Nisitani et al., namely that d(2c)/dN= C(aa/a,)".(2c), where C is a constant and C T~ is the ultimate tensile strength. It is pointed out that an improvement in fatigue strength of cast aluminium alloys can be expected by refining the eutectic silicon rather than by an increase in static strength. NOMENCLATUREare&, = maximum projection area of silicon-particle =maximum projection area of microporosity C,, n = material constants 2c = surface crack length 2c,, 2c, = initial and final crack length d(2c)ldN =crack growth rate F ( x ) = cumulative probability K,__ = maximum stress intensity factor Nf = number of cycles to failure Ni = number of cycles to crack initiation Np = crack growth life So = standard inspection area Sr = ratio of the sum of crack lengths propagated within the cluster of Si-particles to the total crack length Sro = Sr when crack growth is independent of microstructure and is along a straight path T(x) = return period ua = stress amplitude CT, = ultimate tensile strength u0.* = 0.2% offset yield strength