“…It is worth mentioning here that when the fluid domain is unbounded and the velocity u 0 is irrotational (i.e., curl u 0 = 0, a condition that preserved by the evolution), this problem is called the (incompressible) water wave problem, which has received a great deal of attention. The local wellposedness (LWP) for the free-boundary incompressible Euler equations in either bounded or unbounded domains have been studied in [1,4,5,6,9,10,13,24,32,34,36,39,43,44,45,46,47,52,54,55,56,59,60]. In addition, the long time well-posedness for the water wave problem with small initial data is available in [2,16,23,25,27,53,57,58,61], and there are recent results concerning the life-span for the water wave problem with vorticity [17,26,48].…”