RIMS workshop “Nonlinear and Random Waves”
October 3-5, 2022, Kyoto, Japan

Program and Abstract
Oct. 3 2022
9:00-10:00 Andrea Nahmod, University of Massachusetts Amherst“Gibbs measures and propagation of randomness under the flow of nonlinear dispersive PDE”
10:20-11:20 Mamoru Okamoto, Osaka University“Stochastic quantization of the Φ33 model”
11:20-13:50 Lunch
13:50-14:50 Itsuko Hashimoto, Kanazawa University“Asymptotic behavior toward radially symmetric stationary solutions of the compressible Navier-Stokesequation”
15:00-15:30 (Short communication) Hirotatsu Nagoji, Kyoto University“Renormalization of stochastic nonlinear heat and wave equations driven by subordinate cylindrical Brownian noises”
15:30-16:00 Coffee break
16:00-17:00 Laurent Thomann, Universite de Lorraine“Almost sure scattering for the one dimensional nonlinear Schrödinger equation”
17:00-18:00 (HYD2 Joint meeting with RIMS) Gennady EI, Northumbria University“Emergent hydrodynamics of soliton gases in integrable systems”

Oct. 4 2022
9:00-10:00 Sandra Cerrai, University of Maryland“On the small mass limit for some stochastic wave equations with variable friction and polynomial nonlinearities”
10:10-11:10 Masaya Maeda, Chiba University“Small energy stabilization of 1D nonlinear Klein-Gordon equation with potential”
11:20-12:20 Yukimi Goto, Kyushu University“Born-Oppenheimer potential energy surfaces for Kohn-Sham models in the local density approximation
12:20-14:30 Lunch
14:30-15:30 Aurelien Deya, Universite de Lorraine“A few results about the hyperbolic Anderson model”
15:30-16:00 Coffee break
16:00-17:00 Anne-Sophie de Suzzoni, Ecole Polytechnique12“General remarks on the propagation of chaos in wave turbulence and application to the incompressible
Euler dynamics”
17:00-18:00 (HYD2 Joint meeting with RIMS) Christof Melcher, RWTH Aachen University“Emergent spin-orbit coupling and rotating skyrmions”
Oct. 5 2022
9:30-10:30 Takiko Sasaki, Musashino University“The combined effect of one space dimension beyond the general theory for nonlinear wave equations”
10:40-11:10 (Short communication) Minami Watanabe, Tsuda College“Global dynamics of nonlinear Schrödinger equation with double power nonlinearity”
11:20-12:20 Svetlana Roudenko, Florida International University“Dynamics of solutions in stochastic nonlinear Schrödinger equation: critical and supercritical cases”

3

Andrea Nahmod, University of Massachusetts Amherst
Title: Gibbs measures and propagation of randomness under the flow of nonlinear dispersive
PDE.

Abstract: In groundbreaking work, Bourgain ‘96 put forward a random data theory to study theexistence of strong solutions on the statistical ensemble of Gibbs measures associated with dispersive
equations. Despite numerous follow-up works to those of Bourgain’s, fundamental questions remained
open. How does a given initial random data get transported by a nonlinear flow? If it is Gaussian
initially, how does this Gaussianity propagate? What is the description of the solution beyond the linearevolution? In work joint with Yu Deng and Haitian Yue, we developed the theory of random tensors, apowerful new framework which allows us to unravel the propagation of randomness under the NLS flow
beyond the linear evolution of random data, and answer these questions in an optimal range relative towhat we define as the probabilistic scaling. In particular, we establish the invariance of Gibbs measuresfor 2D NLS and 3D Hartree NLS equations using the method of random averaging operators, a firstorder approximation to the full random tensor theory. In this talk we will describe these results, andexplain the ideas behind them. We conclude with some open problems and within this context we’ll
briefly discuss recent joint with Bjoern Bringmann, Yu Deng and Haitian Yue establishing the invarianceof the 3D Gibbs measure under the flow of the nonlinear wave equation.

Mamoru Okamoto, Osaka University
Title: Stochastic quantization of the Φ33 model

Abstract: We study the construction of the Φ33 -measure and the corresponding dynamical problem. We
prove the normalizability of the Φ33 -measure, which is singular with respect to the massive Gaussian freefield. We also consider the dynamical problem for the canonical stochastic quantization of the Φ33 -measure,namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity
forced by an additive space-time white noise. This is a joint work with Tadahiro Oh (University ofEdinburgh) and Leonardo Tolomeo (University of Bonn).

Itsuko Hashimoto, Kanazawa University
Title: Asymptotic behavior toward radially symmetric stationary solutions of the compressible Navier-Stokes equation
Abstract: This talk is concerned with the asymptotic behaviors of radially symmetric stationary
solutions for the compressible Navier-Stokes equation. We discuss the motion of viscous barotropic gas
without external forces, where boundary and far field data are prescribed on the exterior domain. For
both inflow and outflow problems, the asymptotic behaviors of radially symmetric stationary solutions
are shown in a suitably small neighborhood of the initial data. This research is the joint work withProfessor Nishibata and Sugisaki of Tokyo Institute of Technology and Matsumura of Osaka university.4Hirotatsu Nagoji, Kyoto University
Title: Renormalization of stochastic nonlinear heat and wave equations driven by subordinate cylindrical Brownian noises

Abstract: In this talk, we consider the stochastic nonlinear heat equations (SNLH) and stochastic
nonlinear wave equations (SNLW) on two-dimensional torus driven by a subordinate cylindrical Brownian noise, which we define by the time-derivative of a cylindrical Brownian motion subordinated to anondecreasing cadlag stochastic process. To construct the solution, we introduce a suitable renormalization. For SNLH, we cannot expect the time-continuity for the solutions because the noise is jump-type.
Moreover, due to the low time-integrability of the solutions, I could establish a local well-posedness result for SNLH only with a quadratic nonlinearity. On the other hand, for SNLW, the solutions havetime-continuity and we can show the local well-posedness for general polynomial nonlinearities. Throughthis example, we can see that the heat case behaves worse than the wave case in the singular noise ofjump-type cases.

Laurent Thomann, Universite de Lorraine
Title : Almost sure scattering for the one dimensional nonlinear Schrödinger equation

Abstract : We exhibit measures on the space of initial data for which we describe the non trivial
evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on theRadon-Nikodym derivatives of these measures with respect to each other and we characterise their Lpregularity. We deduce from this precise description the global well-posedness of the equation for p > 1
and scattering for p > 3. This is joint work with Nicolas Burq.

(HYD2 Joint meeting with RIMS) Gennady EI, Northumbria University
Title: Emergent hydrodynamics of soliton gases in integrable systems

Abstract: Soliton gases represent infinite random ensembles of interacting solitons displaying nontrivial
large-scale behaviours ultimately determined by the properties of the elementary two-soliton collisions.
The emergent hydrodynamics of non-equilibrium soliton gases in integrable dispersive systems such as theKorteweg-de Vries and nonlinear Schrödinger equations is described by the universal nonlinear integrodifferential kinetic equation for the density of states in the spectral (Lax) phase space. In my talk, I will
outline the main ideas of the spectral theory of soliton gases and its connection with the fundamental
concept of “integrable turbulence” introduced by Zakharov in 2009.

Sandra Cerrai, University of Maryland
Title: On the small mass limit for some stochastic wave equations with variable friction
and polynomial nonlinearities.
Abstract: We study the validity of the Smoluchowski-Kramers approximation for a class of stochastic
nonlinear damped wave equations, including equations of Klein-Gordon type. We will also study thevalidity of a large deviation principle, in the joint small mass and small noise limit. The friction term isassumed to be state dependent.

5Masaya Maeda, Chiba University
Title: Small energy stabilization of 1D nonlinear Klein-Gordon equation with potential
Abstract: This talk is based on the joint work with S.Cuccagna and S.Scrobogna. We give a partial
extension to dimension 1 of the result proved by Bambusi and Cuccagna on the absence of small energy
real valued periodic solutions for the nonlinear Klein Gordon equation in dimension 3. We combine theframework in Kowalczyk and Martel with the notion of "refined profile.”
Yukimi Goto, Kyushu University
Title: Born-Oppenheimer Potential Energy Surfaces for Kohn-Sham Models in the Local
Density Approximation
Abstract : We consider a system of two electrically neutral (equal number of protons and electrons)
atoms. In quantum mechanics textbooks, these atoms attract each other with a long-range force called
the van der Waals force. This fact has been rigorously proved by Lieb and Thirring in non-relativistic
Schroedinger theory (Phys. Rev. A 1986). However, the repulsive force should be stronger for small
nuclear separation. In this talk, I will explain that this repulsive force is about the -7th power of thedistance R, using a nonlinear equation called the local density approximation based on the density matrix
functional theory.
Aurelien Deya, Universite de Lorraine
Title: A few results about the hyperbolic Anderson model
Abstract: We will report on recent results about the following hyperbolic Anderson model:∂2u(t, x) = ∆u(t, x) + u Ḃ(t, x),∂t2t ∈ [0, T ], x ∈ Rd , d ∈ {1, 2, 3},

where Ḃ is a space-time fractional noise of index H = (H0 , ..., Hd ) ∈ (0, 1)d+1 , that is a Gaussian noise
on R+ × Rd with covariance (formally) given bydY|xi − yi |2Hi −2 .
E Ḃ(s, x)Ḃ(t, y) = |s − t|2H0 −2i=1

Fractional noises are known to model long-range-dependence phenomena, in contrast with the standard
white-noise situation. Modifying the index H also allows us to calibrate the roughness of the noise.
As in the case of the fractional SDE, the above equation can be investigated along two directions:(1) The stochastic approach, based on the Wick product and the related Skorohod integral.
(2) The pathwise approach, based on the construction of the Young (wave) integral.
We will discuss about these two possible interpretations, and account for the corresponding results on
wellposedness of the model.
The talk will be based on two joint works with Xia Chen, Jian Song and Samy Tindel.
Anne-Sophie de Suzzoni, Ecole PolytechniqueTitle : General remarks on the propagation of chaos in wave turbulence and application to
the incompressible Euler dynamics
Abstract : In this talk, we will present a result regarding propagation of chaos in the context of wave
turbulence. That is, considering the solution to Hamiltonian PDEs having at initial time independent
Gaussians as Fourier coefficients, we prove that at later times, the Fourier coefficients remain asymptotically independent Gaussians in the sense that they satisfy the Wick formula. The Wick formula at latertimes is asymptotic in the sense that we work on a torus whose size grows to infinity. We then apply thisresult to the incompressible Euler equation.

6(HYD2 Joint meeting with RIMS) Christof Melcher, RWTH Aachen University
Title: Emergent spin-orbit coupling and rotating skyrmions
Abstract: We discuss solitonic field configurations on a spherical magnet. Exploiting the Hamiltonian
structure and concepts of angular momentum, we present a new family of localized solutions to theLandau-Lifshitz equation that are topologically distinct from the ground state and break rotational
symmetry. The approach illustrates emergent spin-orbit coupling arising from the loss of individual
rotational invariance in spin and coordinate space - a common feature of condensed matter systems with
topological phases.

Takiko Sasaki, Musashino University/Tohoku University
Title: The combined effect of one space dimension beyond the general theory for nonlinear
wave equations
Abstract: In this talk, we introduce lifespan estimates for a special class of semilinear wave equations in
one space dimension for which the so-called combined effect is observed. It is remarkable that, including
the combined effect case, our results on the lifespan estimates are partially better than those of thegeneral theory for nonlinear wave equations. This is a joint work with Professor Hiroyuki Takamura andMr. Katsuaki Morisawa.

Minami Watanabe, Tsuda College
Title: Global dynamics of nonlinear Schrödinger equation with double power nonlinearity
Abstract: In this presentation, we consider the global behavior of solutions to the Schrödinger equation(NLS) with double-power nonlinear terms. The Schrödinger equation, which is classified as a nonlinear
dispersive equation, is classified into a scattering solution, an explosive solution, and a standing wavesolution, depending on the competition between wave dispersiveness and nonlinearity. The goal is toclassify the above three solutions by the initial values. Therefore, we set up potential wells for positive
and negative virial functionals, and report the results of investigating the solutions of (NLS) starting fromeach set. We also touch on recent results on the global behavior of solutions to the nonlinear Schrödinger
equation.

Svetlana Roudenko, Florida International University
Title: Dynamics of solutions in stochastic nonlinear Schrödinger equation: critical andsupercritical cases
Abstract: We consider the one-dimensional focusing stochastic nonlinear Schrödinger (SNLS) equation with linear and nonlinear noise in the L2 -critical and supercritical settings. We look at additiveand multiplicative perturbations driven by space-time white noise and multiplicative noise driven by a
Wiener process white in time and colored in space. We discuss the influence of noise on conserved (indeterministic setting) quantities, and then on the dynamics of solutions. We discuss various thresholds
for solutions behavior and especially investigate the noise influence onto perturbations of solitons. We
discuss quantitative estimates and also numerical findings. The talk is based on joint work with Annie
Millet, Alex D. Rodriguez and Kai Yang.