表紙・目次

概要

This publication is the proceedings of the hybrid format workshop entitled "Nonlinear and Random
Waves", which was held during October 3-5, 2022, at the Research Institute for Mathematical Sciences
(RIMS), Kyoto University, Japan.
The main scientific subject of the workshop was nonlinear singular stochastic dispersive equations.
Studies of nonlinear wave propagation in a random environment, or forced by a stochastic perturbation
arc of great importance in engineering and physics: nonlinear optics, condensed matter physics, fluid
mechanics, turbulence analysis. Since Martin Hairer's contribution in the field of stochastic PDEs,
singular stochastic parabolic equations were very competitive research subjects these last five years, but
applications of Hairer's theory are limited for nonlinear singular dispersive equations due to the lack of
smoothing properties, although the wave equations can be accessible somehow by case. On the other
hand, Bourgain's almost-everywhere approach by the use of Gibbs measure was followed by numerous
developements, and simultaneously the study of the propagation of randomness under Hamiltonian flows,
like wave equations, attracts now many researchers in the world. And both topics are closely related.
The purpose of this workshop was to broaden such arguments for random nonlinear dispersive equations
from different and various aspects, for example, the propagation of randomness under nonlinear dispersive
equations, asymptotic properties like large deviation principle and scaling limits, qualitative properties
of nonlinear dispersive-type equations, scattering and stability of special solutions.
The content of this proceedings consists of short notes by Anne-Sophie de Suzzoni, Aurelien Deya,
Hirotatsu Nagoji, Itsuko Hashimoto, Laurent Thomann, Mamoru Okamoto, Masaya Maeda, Minami
Watanabe and Yukimi Goto. Those notes are related to the topics of their talks in the workshop and
cast a spotlight on recent developpements in nonlinear and random wave problems : The note by A-S.
de Suzzoni is concerned with the propagation of chaos in wave turbulence modeled by the incompressible
Euler equation on the torus with size L, and explains in which sense a solution corresponding to a random
initial data with independent Gaussian Fourier coefficients sees its Fourier coefficients at a later fixed
time remain independent for sufficiently large L. The note by A. Deya summarizes the well-posedness
issues for the linear wave equation driven by a multiplicative space-time fractional noise, proposing two
approaches to give sense to the stochastic product: Skorohod approach and pathwise approach. H. Nagoji
offers in his note a new result as an application of the I-method for the global existence of solutions for
two-dimensional nonlinear wave equations driven by subordinate cylindrical noises, following the local
existence result given in his talk. I. Hashimoto explains her result on the asymptotic stability of the
stationary solution on the outflow problem described by the compressible Navier-Stokes equation in the
exterior domain of a ball. L. Thomann gives an overview of his results on the almost sure global existence
and the almost sure scattering of the solution of the one-dimensional nonlinear Schrodinger equation in
the whole space with random initial data. The new idea is the use of the absolute continuity between the
linear flow measure and the nonlinear flow measure, originated from the quasi-invariance of the initial
measure in the Hamiltonian structure, and the monotonicity of the Gibbs type measure. This method
seems more applicable to various situations than Bourgain's original arguments since it does not make use
of the 'invariance' of the measure. M. Okamoto, in his note, focuses on the phenomenon of phase transition
in terms of non normalizability/normalizability for ~-measure and Gibbs measure under Hartree type
interactions. M. Maeda's note, as a brief resume of the works by Kowalczyk, Martel, Mufi.oz, Van den
Bosch, introduces a virial method to prove the decay estimate of the solution of Schrodinger equations,
and develop some ideas on how to remove the eigenvalues which are an obstacle for such decay properties.
M. Watanabe considers the classification problem in the nonlinear Schrodinger equation, in case of double
power nonlinearities. ...