ProbDynDispEq SIGNED
Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
Total Cost €
0EC-Contrib. €
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Project "ProbDynDispEq" data sheet
The following table provides information about the project.
| Coordinator |
THE UNIVERSITY OF EDINBURGH
Organization address contact info |
| 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.
| # | ||||
|---|---|---|---|---|
| 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 |
|---|---|---|---|
| 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 |
Ãrpád 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 |
Ãrpád 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 |
Ãrpád 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 Vişan 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 |
Ãrpád 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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