Strichartz estimates and local existence for the gravity–capillary waves with non-Lipschitz initial velocity

We consider the gravity–capillary waves in any dimension and in fluid domains with general bottoms. Using the paradifferential reduction established in [19], we prove Strichartz estimates for solutions to this problem, at a low regularity level such that initially, the velocity field can be non-Lipschitz up to the free surface. We then use those estimates to solve the Cauchy problem at this level of regularity.