Titles & Abstracts
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Speaker: Yvan Martel (UVSQ - Université Paris-Saclay and Institut Universitaire de France)
Title: Non-existence of internal mode for the 1D Zakharov system
Abstract:
We prove that the linearized operator around any sufficiently small solitary wave of the one-dimensional Zakharov system has no internal mode. This spectral result is expected to play a role in any attempt to study the asymptotic stability of small solitary waves for this model.
Work in collaboration with Guillaume Rialland (UVSQ, France).
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Speaker: Michael McNulty (Universität zu Köln)
Title: Stable blowup for the higher-dimensional Skyrme model
Abstract:
The Skyrme model is a geometric field theory and a quasilinear modification of the nonlinear sigma model, i.e., wave maps into the sphere. In 3+1 dimensions, it is known that this modification prevents self-similar collapse and that global existence holds for large data. I will discuss recent work which demonstrates that this is not the case in 5+1 dimensions. In particular, Skyrme’s modification provides a stable mechanism for finite-time blowup at the self-similar rate.
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Speaker: Ruoyuan Liu (Universität Bonn)
Title: Global well-posedness of the two-dimensional dispersive Anderson model and its large torus limit
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 divided into two parts.
In the first part, we study global well-posedness of the DAM on the
plane. We proceed by using a gauge transform introduced by Hairer and
Labbé (2015) on the parabolic Anderson model and construct 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. Due to 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 fully exploit the
quantitative convergence of the periodic noise to the full space noise
via function spaces with exponential weights. This part is based on a
joint work with Nikolay Tzvetkov (ENS Lyon).
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