Global well-posedness for the two-dimensional dispersive Anderson model and its large torus limit
A talk in the Oberseminar Analysis series by
Ruoyuan Liu from Bonn
| Abstract: | In this talk, we consider the two-dimensional nonlinear Schrödinger
equation with a multiplicative spatial white noise and a polynomial
nonlinearity, also known as the dispersive Anderson model (DAM). The
talk is split into two parts.
In the first part, we study global well-posedness for the DAM on the
plane. We proceed by using a gauge transform introduced by Hairer and
Labbé (2015) on the parabolic Anderson model and constructing the
solution as a limit of solutions to a family of approximating
equations. To establish global well-posedness, we establish a priori
bounds using the Hamiltonian structure of the equation and also
Strichartz estimates. In order to control the logarithmic growth of
the noise, we incorporate function spaces with polynomial weights in
our analysis. This part is based on a joint work with Arnaud Debussche
(ENS Rennes), Nikolay Tzvetkov (ENS Lyon), and Nicola Visciglia
(University of Pisa).
In the second part, we show that the global solution of the DAM on the
plane can be realized as a limit of the periodic global dynamics of
the DAM as the period goes to infinity, given suitable initial
conditions and periodization of the noise. Global well-posedness of
the periodic DAM was shown by Tzvetkov and Visciglia (2023), but the
global solutions are not uniform in periods due to the logarithmic
growth of the noise. To overcome this issue, we introduce periodic
weights and construct weighted function spaces on periodic domains,
which allow us to obtain a priori bounds for solutions independent of
the periodicity. This part is based on a joint work with Nikolay
Tzvetkov (ENS Lyon). Zoom-Meeting beitreten uni-bielefeld.zoom-x.de/j/64951709096?pwd=EfJynCkou8bBw77CsNYMuh88IrKazJ.1 Meeting-ID: 649 5170 9096 Passwort: 351532 Within the CRC this talk is associated to the project(s): A1 |