Propagation of long-crested water waves

Propagation of long-crested water waves. Ⅱ. Bore propagation

“…The present work allows for much stronger variations in the y-directions, albeit still maintaining the long-crested hypothesis that the motion becomes two-dimensional as y → ±∞. As mentioned, local well-posedness for exactly this problem is in hand (see [9]).…”

“…Missing from our previous analysis was theory that extended to the Boussinesq time scale 1 ε . Such a result was obtained in case η 1 = 0 under regularity assumptions that imply the initial data is localized in the x-directions (see again [9]). This longer-time theory does not apply when η 1 > 0.…”

“…There is a substantial theory pertaining to the initial-value problem for these systems on the Boussinesq time scale O( 1 ε ) when the initial data is assumed to decay to zero (see [2], [20], [21], [26] for the long-crested situation where variations in the y-directions are ignored and [16] when the waves are three-dimensional). Work on bore-type problems for two-dimensional Boussinesq systems appeared in [9] and in the recent independent, but related work of Burtea [15].…”

“…We remark that in our earlier paper [9], we took u 1 = u 2 = 0, corresponding to completely quiescent water. In the present study, this is relaxed.…”

“…In the present study, this is relaxed. It is staightforward to check that all the major results in [9] go over for this slightly more general situation. Indeed, all that is required is to substract from u 0 the function χ to be introduced presently and work with this new dependent variable.…”