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

Abstract

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