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We provide a complete local well-posedness theory in $H^s$ based Sobolevspaces for the free boundary incompressible Euler equations with zero surfacetension on a connected fluid domain. Our well-posedness theory includes: (i)Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness,and the first proof of continuous dependence on the data, all in low regularitySobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at thelevel of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$regularity of the free surface; (iii) Stability bounds: We construct anonlinear functional which measures, in a suitable sense, the distance betweentwo solutions (even when defined on different domains) and we show that thisdistance is propagated by the flow; (iv) Energy estimates: We prove refined,essentially scale invariant energy estimates for solutions, relying on a newlyconstructed family of elliptic estimates; (v) Continuation criterion: We givethe first proof of a sharp continuation criterion in the physically relevantpointwise norms, at the level of scaling. In essence, we show that solutionscan be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the freesurface is in $L_T^1C^{1,\frac{1}{2}}$, which is at the same level as theBeale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of theconstruction of regular solutions.  Our entire approach is in the Eulerian framework and can be adapted to workin more general fluid domains.

Sharp Hadamard local well-posedness, enhanced uniqueness and pointwise continuation criterion for the incompressible free boundary Euler equations