Bi-parameter paracontrolled approach to singular stochastic wave equations

奇异随机波动方程的双参数参数控制方法

基本信息

  • 批准号:
    EP/Y033507/1
  • 负责人:
  • 金额:
    $ 4.54万
  • 依托单位:
  • 依托单位国家:
    英国
  • 项目类别:
    Research Grant
  • 财政年份:
    2024
  • 资助国家:
    英国
  • 起止时间:
    2024 至 无数据
  • 项目状态:
    未结题

项目摘要

Nonlinear dispersive partial differential equations (PDEs), such as the nonlinear wave equations (NLW), appear ubiquitously as models describing wave propagation in various branches of physics and engineering. In particular, it is well known that the one-dimensional wave equation describes the motion of a vibrating string. In a physical setting, such a vibrating string is susceptible to external forcing which is often random. Such a random external forcing is well approximated by a white noise in many situations. For this reason, it is of fundamental physical importance to study the stochastic NLW forced by space-time white noise. At the same time, such a problem also poses significant analytical challenges due to the irregularity of the space-time white noise. The main aim of this proposal is to advance our theoretical understanding of the one-dimensional stochastic NLW with multiplicative space-time white noise forcing by working on concrete examples of challenging open problems.The main difficulty of mathematical analysis on singular stochastic PDEs with white noise forcing comes from the irregularity of the white noise. Over the last decade, we have seen a tremendous progress in the study of singular stochastic PDEs with white noise forcing. In the parabolic setting, this development was led by a 2014 Fields medalist, Hairer (EPFL, Switzerland), and by Gubinelli (Oxford). Over the last five years, the principal investigator (PI) has made a substantial, world-leading contribution to the development of our theoretical understanding of singular stochastic NLW.In the proposed projects, the PI will study several important models of one-dimensional stochastic NLW with multiplicative space-time white noise forcing and aims to resolve challenging open problems by establishing their pathwise well-posedness. The PI plans to achieve this goal by developing an entirely new analytical framework which allows him to handle singular multiplicative noises in a pathwise manner.
非线性色散偏微分方程(PDE),如非线性波动方程(NLW),作为描述波传播的模型在物理和工程的各个分支中普遍存在。特别是,众所周知,一维波动方程描述了振动弦的运动。在物理环境中,这种振动的弦容易受到外部力量的影响,而外部力量通常是随机的。在许多情况下,这样的随机外力可以用白色噪声很好地近似。因此,研究时空白色噪声作用下的随机NLW具有重要的物理意义。同时,由于时空白色噪声的不规则性,这样的问题也带来了重大的分析挑战。本文的主要目的是通过具体的开放性问题的实例来推进我们对具有乘性时空白色噪声强迫的一维随机NLW问题的理论理解.具有白色噪声强迫的奇异随机偏微分方程的数学分析的主要困难来自于白色噪声的不规则性.在过去的十年中,我们看到了白色噪声强迫下奇异随机偏微分方程的研究取得了巨大的进展。在抛物线设置中,这一发展由2014年菲尔兹奖得主Hairer(瑞士洛桑联邦理工学院)和Gubinelli(牛津大学)领导。在过去的五年中,首席研究员(PI)作出了重大的,世界领先的贡献,我们的奇异随机NLW的理论理解的发展。在拟议的项目中,PI将研究几个重要的模型,一维随机NLW乘性时空白色噪声强迫,旨在解决具有挑战性的开放问题,通过建立他们的路径适定性。PI计划通过开发一个全新的分析框架来实现这一目标,该框架允许他以路径方式处理奇异乘性噪声。

项目成果

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Tadahiro (Choonghong) Oh其他文献

Tadahiro (Choonghong) Oh的其他文献

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