Shao Liu - Probabilistic well-posedness of dispersive PDEs beyond variance blowup
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UMPA, ENS de Lyon
In this talk, we investigate a possible extension of
probabilistic well-posedness theory of nonlinear dispersive PDEs with
random initial data beyond variance blowup. As a model equation, we
consider the quadratic nonlinear wave equation (qNLW) on the
two-dimensional torus with (rough) Gaussian initial data. By introducing a
suitable vanishing multiplicative renormalization constant on the initial
data, we show that solutions to qNLW with the renormalized (rough)
Gaussian initial data converge to a solution to the stochastic qNLW forced
by a space-time noise. This work is a continuation of recent work (2025)
by G. Li, J. Li, T. Oh, and N. Tzvetkov, in which they initiated this line of
research by studying the Benjamin-Bona-Mahony equation. In their setting,
dispersion does not play a crucial role, whereas it is a key ingredient in
our analysis. This talk is based on a joint work with Guopeng Li (BIT),
Jiawei Li (Edinburgh), Tadahiro Oh (Edinburgh), and Nikolay Tzvetkov (ENS
Lyon).