Dynamics of singular stochastic nonlinear dispersive PDEs

奇异随机非线性色散偏微分方程的动力学

• 批准号: EP/V003178/1
• 负责人: Yuzhao Wang
• 金额: $ 33.03万
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV003178%2F1
• 关键词: Dynamics; singular; stochastic; nonlinear; dispersive

Dispersion exists ubiquitously in nature. The most famous example of dispersion is seen in a rainbow, where dispersion effect separates the white light spatially into components of different wavelengths (different colours). Nonlinear dispersive partial differential equations (PDEs), such as nonlinear Schrodinger equations (NLS) and nonlinear wave equations (NLW), appear naturally in models describing wave phenomena in the real world. In the past thirty years, the study of deterministic nonlinear dispersive PDEs has seen significant development, in which harmonic analysis has played a fundamental role, led by Kenig, Bourgain and Tao, among others. In recent years, a combination of deterministic analysis with probability theory has played an increasingly important role in the field. This probabilistic perspective allows us to go beyond the limits of deterministic analysis. More importantly, it is also essential to understand the effect of stochastic perturbation in practice since such stochastic perturbation is ubiquitous.The main objective of this research is to develop novel mathematical ideas and techniques to clarify long-standing fundamental questions in the study of stochastic nonlinear dispersive PDEs, with primary examples given by stochastic NLS and stochastic NLW. In the field of singular stochastic parabolic PDEs, significant progress has been taking place led by Hairer and Gubinelli with their collaborators. This has enabled striking theories which are changing the landscape of the study in this field. However, their new theories are designed to handle parabolic problems, and it is not a priori clear on how to adapt them to solve dispersive equations. Despite some exciting recent progress, our understanding of stochastic dispersive PDEs is still very far from satisfactory. In these proposed projects, the principal investigator (PI) will study several open problems in the field of stochastic dispersive PDEs. More specifically, the PI will focus on studying the properties of invariant measures and the local and global-in-time solutions to stochastic NLS and NLW in periodic domains. The PI plans to address these problems by combining tools from dispersive PDEs, stochastic analysis, probability theory and harmonic analysis with recent progress.

分散现象在自然界中普遍存在。色散最著名的例子是在彩虹中看到的，其中色散效应将白色光在空间上分离成不同波长（不同颜色）的分量。非线性色散偏微分方程，如非线性薛定谔方程（NLS）和非线性波动方程（NLW），自然出现在描述真实的世界中波动现象的模型中。在过去的三十年里，确定性非线性色散偏微分方程的研究取得了重大的发展，其中调和分析在其中扮演了重要的角色，主要由Kenig，Bourgain和Tao等人领导。近年来，确定性分析与概率论的结合在该领域发挥着越来越重要的作用。这种概率观点使我们能够超越确定性分析的限制。更重要的是，它也是必要的，以了解在实践中的随机扰动的影响，因为这样的随机扰动是无处不在的。本研究的主要目标是发展新的数学思想和技术，以澄清长期存在的基本问题，在随机非线性色散偏微分方程的研究，随机NLS和随机NLW给出的主要例子。在奇异随机抛物偏微分方程领域，Hairer和Gubinelli及其合作者取得了重大进展。这使得引人注目的理论正在改变这一领域的研究景观。然而，他们的新理论是为了处理抛物问题而设计的，如何使它们适用于求解色散方程并不是先验清楚的。尽管最近取得了一些令人兴奋的进展，我们对随机色散偏微分方程的理解仍然远远不能令人满意。在这些项目中，主要研究者（PI）将研究随机色散偏微分方程领域的几个开放性问题。更具体地说，PI将专注于研究不变测度的性质以及周期域中随机NLS和NLW的局部和全局时间解。PI计划通过将分散偏微分方程、随机分析、概率论和调和分析的工具与最新进展相结合来解决这些问题。

项目成果

Gibbs Measure for the Focusing Fractional NLS on the Torus

圆环上聚焦分数 NLS 的吉布斯测量

• DOI: 10.1137/21m1445946
• 发表时间: 2022
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Liang R

On the Parabolic and Hyperbolic Liouville Equations

• DOI: 10.1007/s00220-021-04125-8
• 发表时间: 2019-08
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Tadahiro Oh;T. Robert;Yuzhao Wang

Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

• DOI: 10.1007/s40072-022-00237-x
• 发表时间: 2021-06
• 期刊: Stochastics and Partial Differential Equations: Analysis and Computations
• 作者: Tadahiro Oh;Yuzhao Wang;Younes Zine

Improved bilinear Strichartz estimates with application to the well-posedness of periodic generalized KdV type equations

改进的双线性 Strichartz 估计并应用于周期性广义 KdV 型方程的适定性

• DOI: 10.48550/arxiv.2207.08725
• 发表时间: 2022
• 作者: Molinet L