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We consider the three-dimensional Euler equations in a domain with a freeboundary with no surface tension. We construct unique local-in-time solutionsin the Lagrangian setting for $u_0 \in H^{2.5+\delta }$ such that theRayleigh-Taylor condition holds and $\mathrm{curl}\,u_0 \in H^{2+\delta }$ inan arbitrarily small neighborhood of the free boundary. We show that the resultis optimal in the sense that $H^{3+\delta}$ regularity of the Lagrangiandeformation near the free boundary can be ensured if and only if initialvorticity has $H^{2+\delta}$ regularity of vorticity near the free boundary.

Construction of the free-boundary 3D incompressible Euler flow under limited regularity