Low regularity solutions for gravity water waves

We prove local well-posedness for the gravity water waves equations without surface tension, with initial velocity field in $$H^s$$Hs, $$s > \frac{d}{2} + 1 - \mu $$s>d2+1-μ, where $$\mu = \frac{1}{10}$$μ=110 in the case $$d = 1$$d=1 and $$\mu = \frac{1}{5}$$μ=15 in the case $$d \ge 2$$d≥2, extending previous results of Alazard–Burq–Zuily. The improvement primarily arises in two areas. First, we perform an improved analysis of the regularity of the change of variables from Eulerian to Lagrangian coordinates. Second, we perform a time-interval length optimization of the localized Strichartz estimates.