The APDE seminar on Monday, 9/25, will be given by Zhongkai Tao (Berkeley) in-person in **Evans 736,** and will also be broadcasted online via Zoom from **4:10pm to 5:00pm PST**. To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

**Title: **Sharp Hadamard well-posedness for the incompressible free boundary Euler equations

**Abstract:** I will talk about a recent preprint in which we establish an optimal local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations on a connected fluid domain. Some components of this result include: (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 regularity Sobolev spaces; (ii) Enhanced uniqueness: A uniqueness result which holds at the level of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove essentially scale invariant energy estimates for solutions, relying on a newly constructed family of refined elliptic estimates; (v) Continuation criterion: We give the first proof of a continuation criterion at the same scale as the classical Beale-Kato-Majda criterion for the Euler equation on the whole space. Roughly speaking, we show that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of the construction of regular solutions.

Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is based on joint work with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.