Seminar

We provide a complete local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension 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 regularity Sobolev spaces; (ii) Enhanced uniqueness 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); (iv) Essentially scale invariant energy estimates for solutions, relying on a newly

constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a breakdown criterion in pointwise norms at the same scale as the Beale-Kato-Majda criterion for the Euler equation on the whole space; (vi) A novel proof of the construction of regular solutions. This is joint work with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.

Meeting ID: 998 5718 9855

Passcode: 983468

Link: https://msri.zoom.us/j/99857189855?pwd=LzRFR2tPN1cydWJNZEZkclZGV2lpQT09