Analysis Seminar

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On the wave turbulence theory of 2D gravity waves

Speaker: Alex Ionescu, Princeton University

Location: Warren Weaver Hall 1302

Date: Thursday, March 14, 2024, 11 a.m.

Synopsis:

Our goal in joint work with Yu Deng and Fabio Pusateri is to initiate the rigorous investigation of wave turbulence for water wave models. This problem has received intense attention in recent years in the context of semilinear models, such as semilinear Schrodinger equations or multi-dimensional KdV-type equations. However, our situation is different since water wave systems are quasilinear and the solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss.

Our strategy consists of two main steps: (1) a deterministic energy inequality that provides control of (possibly large) Sobolev norms of solutions for long times, under the condition that a certain $L^\infty$-type norm is small, and (2) a propagation of randomness argument to prove a probabilistic regularity result for long times, in a suitable scaling regime.