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ProbDynDispEq SIGNED

Probabilistic and Dynamical Study of Nonlinear Dispersive Equations

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EC-Contrib. €

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Partnership

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Project "ProbDynDispEq" data sheet

The following table provides information about the project.

Coordinator
THE UNIVERSITY OF EDINBURGH 

Organization address
address: OLD COLLEGE, SOUTH BRIDGE
city: EDINBURGH
postcode: EH8 9YL
website: www.ed.ac.uk

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country United Kingdom [UK]
 Project website http://www.maths.ed.ac.uk/
 Total cost 1˙007˙811 €
 EC max contribution 1˙007˙811 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2014-STG
 Funding Scheme ERC-STG
 Starting year 2015
 Duration (year-month-day) from 2015-03-01   to  2020-02-29

 Partnership

Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    THE UNIVERSITY OF EDINBURGH UK (EDINBURGH) coordinator 1˙007˙811.00

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 Project objective

Nonlinear dispersive partial differential equations (PDEs) appear ubiquitously as models describing wave phenomena in various branches of physics and engineering. Over the last few decades, multilinear harmonic analysis has played a crucial role in the development of the theoretical understanding of the subject. Furthermore, in recent years, a non-deterministic point of view has been incorporated into the study of nonlinear dispersive PDEs, enabling us to study typical behaviour of solutions in a probabilistic manner and go beyond the limit of deterministic analysis.

The main objective of this proposal is to develop novel mathematical ideas and techniques, and make significant progress on some of the central problems related to the nonlinear Schrödinger equations (NLS) and the Korteweg-de Vries equation (KdV) from both the deterministic and probabilistic points of view. In particular, we consider the following long term projects:

1. We will study properties of invariant Gibbs measures for nonlinear Hamiltonian PDEs. One project involves establishing a new connection between the limiting behaviour of the Gibbs measures and the concentration phenomena of finite time blowup solutions. The other project aims to understand the space-time covariance of the Gibbs measures in the weakly nonlinear regime.

2. We will first construct the invariant white noise for the cubic NLS on the circle. Then, we will provide a statistical description of the global-in-time dynamics for the stochastic KdV and stochastic cubic NLS on the circle with additive space-time white noise.

3. We will develop novel analytical techniques and construct the local-in-time dynamics for the cubic NLS on the circle in a low regularity.

4. We will advance the understanding of traveling waves and prove scattering for some energy-critical NLS with non-vanishing boundary conditions.

 Publications

year authors and title journal last update
List of publications.
2019 Tadahiro Oh, Mamoru Okamoto
On the stochastic nonlinear Schrödinger equations at critical regularities
published pages: , ISSN: , DOI:
2020-01-24
2018 Trishen S. Gunaratnam, Tadahiro Oh, Nikolay Tzvetkov, Hendrik Weber
Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions
published pages: , ISSN: , DOI:
2020-01-24
2019 Tadahiro Oh, Tristan Robert, Yuzhao Wang
On the parabolic and hyperbolic Liouville equations
published pages: , ISSN: , DOI:
2020-01-24
2018 Tadahiro Oh, Yuzhao Wang
Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces
published pages: , ISSN: , DOI:
2020-01-24
2018 Tadahiro Oh, Yuzhao Wang
Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces
published pages: , ISSN: , DOI:
Journal d\'Analyse Mathematique 2020-01-24
2018 Gubinelli, Massimiliano; Koch, Herbert; Oh, Tadahiro
Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity
published pages: , ISSN: , DOI:
2020-01-24
2018 Tadahiro Oh, Yuzhao Wang
On global well-posedness of the modified KdV equation in modulation spaces
published pages: , ISSN: , DOI:
2020-01-24
2019 Tadahiro Oh, Tristan Robert, Nikolay Tzvetkov
Stochastic nonlinear wave dynamics on compact surfaces
published pages: , ISSN: , DOI:
2020-01-24
2019 Tadahiro Oh, Tristan Robert, Philippe Sosoe, Yuzhao Wang
On the two-dimensional hyperbolic stochastic sine-Gordon equation
published pages: , ISSN: , DOI:
2020-01-24
2019 Justin Forlano
Almost sure global well posedness for the BBM equation with infinite L^2 initial data
published pages: , ISSN: , DOI:
Discrete & Continuous Dynamical Systems series A 2020-01-24
2018 Leonardo Tolomeo
Unique ergodicity for stochastic hyperbolic equations with additive space-time white noise
published pages: , ISSN: , DOI:
2020-01-24
2020 Árpád Bényi, Tadahiro Oh
Modulation spaces with scaling symmetry
published pages: 496-507, ISSN: 1063-5203, DOI: 10.1016/j.acha.2019.04.005
Applied and Computational Harmonic Analysis 48/1 2020-01-24
2019 Tadahiro Oh, Mamoru Okamoto
Comparing the stochastic nonlinear wave and heat equations: a case study
published pages: , ISSN: , DOI:
2020-01-24
2019 Justin Forlano, William J. Trenberth
On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations
published pages: , ISSN: 0294-1449, DOI: 10.1016/j.anihpc.2019.07.006
Annales de l\'Institut Henri Poincaré C, Analyse non linéaire 2020-01-24
2019 Tadahiro Oh, Mamoru Okamoto, Tristan Robert
A remark on triviality for the two-dimensional stochastic nonlinear wave equation
published pages: , ISSN: , DOI:
2020-01-24
2019 Tadahiro Oh, Oana Pocovnicu, Nikolay Tzvetkov
Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces
published pages: , ISSN: , DOI:
2020-01-24
2015 Tadahiro Oh, Nikolay Tzvetkov
On the transport of Gaussian measures under the flow of Hamiltonian PDEs
published pages: 1-9, ISSN: 2266-0607, DOI: 10.5802/slsedp.84
Séminaire Laurent Schwartz — EDP et applications 2020-01-24
2017 Tadahiro Oh, Nikolay Tzvetkov
Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation
published pages: 1121-1168, ISSN: 0178-8051, DOI: 10.1007/s00440-016-0748-7
Probability Theory and Related Fields 169/3-4 2020-01-24
2017 Massimiliano Gubinelli, Herbert Koch, Tadahiro Oh
Renormalization of the two-dimensional stochastic nonlinear wave equations
published pages: 1, ISSN: 0002-9947, DOI: 10.1090/tran/7452
Transactions of the American Mathematical Society 2020-01-24
2017 Tadahiro Oh, Laurent Thomann
Invariant Gibbs measures for the 2-d defocusing nonlinear wave equations
published pages: , ISSN: , DOI:
Annales de la Faculté des Sciences de Toulouse 2020-01-24
2018 Justin Forlano, Tadahiro Oh, Yuzhao Wang
Stochastic cubic nonlinear Schrödinger equation with almost space-time white noise.
published pages: , ISSN: , DOI:
2020-01-24
2015 Árpád Bényi, Tadahiro Oh, Oana Pocovnicu
On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $mathbb {R}^d$, $d geq 3$
published pages: 1-50, ISSN: 2330-0000, DOI: 10.1090/btran/6
Transactions of the American Mathematical Society, Series B 2/1 2020-01-24
2018 Árpád Bényi, Tadahiro Oh, Oana Pocovnicu
On the probabilistic Cauchy theory for nonlinear dispersive PDEs
published pages: , ISSN: , DOI:
Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis 2020-01-24
2017 Tadahiro Oh
A Remark on Norm Inflation with General Initial Data for the Cubic Nonlinear Schrödinger Equations in Negative Sobolev Spaces
published pages: 259-277, ISSN: 0532-8721, DOI: 10.1619/fesi.60.259
Funkcialaj Ekvacioj 60/2 2020-01-24
2016 Tadahiro Oh, Geordie Richards, Laurent Thomann
On invariant Gibbs measures for the generalized KdV equations
published pages: 133-153, ISSN: 1548-159X, DOI: 10.4310/DPDE.2016.v13.n2.a3
Dynamics of Partial Differential Equations 13/2 2020-01-24
2017 Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu
On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities
published pages: , ISSN: , DOI:
arXiv 2020-01-24
2017 Tadahiro Oh, Philippe Sosoe, Nikolay Tzvetkov
An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation
published pages: , ISSN: , DOI:
arXiv 2020-01-24
2018 Zihua Guo, Tadahiro Oh
Non-Existence of Solutions for the Periodic Cubic NLS below ${L}^{{2}}$
published pages: rnw271, ISSN: 1073-7928, DOI: 10.1093/imrn/rnw271
International Mathematics Research Notices 2018, no. 6 2020-01-24
2016 Jaywan Chung, Zihua Guo, Soonsik Kwon, Tadahiro Oh
Normal form approach to global well-posedness of the quadratic derivative nonlinear Schrödinger equation on the circle
published pages: 1273–1297, ISSN: 0294-1449, DOI: 10.1016/j.anihpc.2016.10.003
Annales de l\'Institut Henri Poincare (C) Non Linear Analysis 34 2020-01-24
2017 Tadahiro Oh, Yoshio Tsutsumi, Nikolay Tzvetkov
Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third order dispersion
published pages: , ISSN: , DOI:
2020-01-24
2017 Rowan Killip, Tadahiro Oh, Oana Pocovnicu, Monica ViÅŸan
Solitons and Scattering for the Cubic–Quintic Nonlinear Schrödinger Equation on $${mathbb{R}^3}$$ R 3
published pages: 469-548, ISSN: 0003-9527, DOI: 10.1007/s00205-017-1109-0
Archive for Rational Mechanics and Analysis 225/1 2020-01-24
2017 Tadahiro Oh, Nikolay Tzvetkov
Quasi-invariant Gaussian measures for the two-dimensional defocusing cubic nonlinear wave equation
published pages: , ISSN: , DOI:
arXiv 2020-01-24
2018 Tadahiro Oh, Nikolay Tzvetkov, Yuzhao Wang
Sovling the 4NLS with white noise initial data
published pages: , ISSN: , DOI:
2020-01-24
2018 Soonsik Kwon, Tadahiro Oh, Haewon Yoon
Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line
published pages: , ISSN: , DOI:
2020-01-24
2017 Árpád Bényi, Tadahiro Oh, Oana Pocovnicu
Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on R^3
published pages: , ISSN: , DOI:
arXiv 2020-01-24
2018 Tadahiro Oh, Laurent Thomann
A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations
published pages: 1-49, ISSN: 2194-0401, DOI: 10.1007/s40072-018-0112-2
Stochastics and Partial Differential Equations: Analysis and Computations 2020-01-24
2018 TADAHIRO OH, YUZHAO WANG
GLOBAL WELL-POSEDNESS OF THE PERIODIC CUBIC FOURTH ORDER NLS IN NEGATIVE SOBOLEV SPACES
published pages: 1-80, ISSN: 2050-5094, DOI: 10.1017/fms.2018.4
Forum of Mathematics, Sigma 6 2020-01-24
2016 Tadahiro Oh, Yuzhao Wang
On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle
published pages: , ISSN: 1221-8421, DOI:
\"Analele Ştiinţifice ale Universităţii \"\"Alexandru Ioan Cuza\'\' din Iaşi - Matematică (Annals of the Alexandru Ioan Cuza University - Mathematics)\" 2020-01-24
2018 Tadahiro Oh, Oana Pocovnicu, Yuzhao Wang)
On the stochastic nonlinear Schrödinger equations with non-smooth additive noise
published pages: , ISSN: , DOI:
2020-01-24

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