Low Regularity Solutions for Gravity Water Waves

The gravity water waves equations are a system of partial differential equations which govern the evolution of the interface between a vacuum and an incompressible, irrotational fluid in the presence of gravity. In the case of two dimensions, these equations model non-breaking waves at the surface of a body of water, such as a lake or ocean, while in one dimension, they model non-breaking waves propagating in a channel.We are concerned with the well-posedness of the Cauchy problem for the gravity water waves equations: We seek to show that given an initial configuration of the vacuum-fluid interface and an initial fluid velocity field beneath the interface, there is a unique solution to the gravity water waves equations which matches the given initial data. In particular, we are concerned with the situation where the initial data has low regularity, corresponding to surface waves which are not necessarily smooth.The classical regularity threshold for the well-posedness of the water waves system requires initial velocity field in $H^s$, with $s > \frac{d}{2} + 1$, and can be obtained by proving standard energy conservation estimates. On the other hand, it has been shown that for dispersive equations (equations describing phenomena which disperse waves of different frequencies), one can lower well-posedness regularity thresholds below that which is attainable by energy conservation alone. This was first realized for the nonlinear wave equation via dispersive estimates known as Strichartz estimates, and was first applied toward the well-posedness of gravity water waves by Alazard-Burq-Zuily. However, this approach was implemented as a partial result, using Strichartz estimates with loss relative to what one expects based on the corresponding linearized model problem. In this dissertation, we prove well-posedness with initial velocity field in $H^s$, $s > \frac{d}{2} + 1 - \mu$, where $\mu = \frac{1}{10}$ in the case $d = 1$ and $\mu = \frac{1}{5}$ in the case $d \geq 2$, extending the previous result of Alazard-Burq-Zuily. In the case of one dimension, using a further refined argument, we establish the well-posedness for $s > \half + 1 - \frac{1}{8}$, corresponding to proving lossless Strichartz estimates. This provides the sharp regularity threshold with respect to the approach of combining Strichartz estimates with energy estimates