Deterministic and probabilistic dynamics of nonlinear dispersive PDEs
非线性色散偏微分方程的确定性和概率动力学
基本信息
- 批准号:EP/S033157/1
- 负责人:
- 金额:$ 29.47万
- 依托单位:
- 依托单位国家:英国
- 项目类别:Research Grant
- 财政年份:2020
- 资助国家:英国
- 起止时间:2020 至 无数据
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Nonlinear dispersive partial differential equations (PDEs) such as the nonlinear wave equations and the nonlinear Schrodinger equations, are time evolution equations modelling wave phenomena. They appear ubiquitously in various branches of physics and engineering such as nonlinear optics, plasma physics, water waves and telecommunication systems. Their validity is widely recognised and supported by numerical and experimental evidences.On the one hand, the mathematical theoretical research of nonlinear dispersive PDEs is important for applied sciences since it has provided solid foundations for the verification and applicability of these models. On the other hand, this theoretical research has proved to be very valuable for mathematics itself. Indeed, over the last thirty years, nonlinear dispersive PDEs have presented very difficult and interesting challenges, motivating the development of many new ideas and techniques in mathematical analysis. One of the sources of richness of nonlinear dispersive PDEs is that each subclass of equations poses its own difficulties, thus requiring the elaboration of specific tools.The aim of this proposal is to explore the dynamics of nonlinear dispersive PDEs using mathematical analysis from both deterministic and probabilistic points of view. In the deterministic setting, this proposal focuses on (i) constructing special solutions to a class of nonlinear Schrodinger equations and (ii) proving the long-time existence of solutions to an equation from plasma physics with non-constant vorticity. The principal investigator (PI) plans to combine PDE techniques with tools from harmonic analysis and spectral theory.In the traditional (deterministic) study of nonlinear evolution equations, one aims to construct solutions to a given PDE for all initial data. In applications, however, one is often content with understanding the behaviour of typical solutions, neglecting rare pathological behaviours. This point of view can be made rigorous by employing probability theory and has led to exciting developments over the last decade. In particular, it has allowed us to go beyond the limits of deterministic analysis. One aspect of this proposal is to investigate dynamics of nonlinear dispersive PDEs from a probabilistic point of view. More specifically, the PI will focus on constructing well-defined dynamics with rough and random initial data by incorporating ideas and tools from probability theory and the very active field of singular stochastic PDEs.
非线性色散偏微分方程(PDE),如非线性波动方程和非线性薛定谔方程,是模拟波动现象的时间演化方程。它们普遍存在于物理学和工程学的各个分支,如非线性光学、等离子体物理、水波和通信系统。一方面,非线性色散偏微分方程的数学理论研究对于应用科学具有重要意义,因为它为这些模型的验证和适用性提供了坚实的基础。另一方面,这一理论研究对数学本身也是很有价值的。事实上,在过去的三十年里,非线性色散偏微分方程提出了非常困难和有趣的挑战,激发了许多新的思想和技术的发展在数学分析。非线性色散偏微分方程的丰富性的来源之一是,每个子类的方程构成了自己的困难,因此需要制定具体的tools.The目的的建议是探索非线性色散偏微分方程的动力学使用数学分析从确定性和概率的观点。在确定性条件下,本文的主要工作是:(i)构造一类非线性Schrodinger方程的特解;(ii)证明一类具有变常涡量的等离子体方程解的长期存在性。主要研究者(PI)计划将联合收割机PDE技术与调和分析和谱理论的工具相结合。在非线性发展方程的传统(确定性)研究中,人们的目标是为所有初始数据构造给定PDE的解。然而,在应用中,人们往往满足于理解典型溶液的行为,而忽略了罕见的病理行为。这一观点可以通过运用概率论来严格化,并在过去十年中取得了令人兴奋的发展。特别是,它使我们能够超越确定性分析的限制。这个建议的一个方面是从概率的角度来研究非线性色散偏微分方程的动力学。更具体地说,PI将专注于通过结合概率论和奇异随机偏微分方程的非常活跃的领域的思想和工具,用粗糙和随机的初始数据构建定义良好的动态。
项目成果
期刊论文数量(10)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces
- DOI:
- 发表时间:2019-04
- 期刊:
- 影响因子:0
- 作者:Tadahiro Oh;Oana Pocovnicu;N. Tzvetkov
- 通讯作者:Tadahiro Oh;Oana Pocovnicu;N. Tzvetkov
Ground state energy threshold and blow-up for NLS with competing nonlinearities
- DOI:10.2422/2036-2145.202005_044
- 发表时间:2020-12
- 期刊:
- 影响因子:0
- 作者:J. Bellazzini;Luigi Forcella;V. Georgiev
- 通讯作者:J. Bellazzini;Luigi Forcella;V. Georgiev
Qualitative Properties of Dispersive PDEs
色散偏微分方程的定性性质
- DOI:10.1007/978-981-19-6434-3_2
- 发表时间:2022
- 期刊:
- 影响因子:0
- 作者:Bellazzini J
- 通讯作者:Bellazzini J
Probabilistic local Cauchy theory of the cubic nonlinear wave equation in negative Sobolev spaces
负Sobolev空间中三次非线性波动方程的概率局部柯西理论
- DOI:10.5802/aif.3454
- 发表时间:2022
- 期刊:
- 影响因子:0
- 作者:Oh T
- 通讯作者:Oh T
Dynamics of solutions to the Gross-Pitaevskii equation describing dipolar Bose-Einstein condensates
- DOI:
- 发表时间:2021-10
- 期刊:
- 影响因子:0
- 作者:J. Bellazzini;Luigi Forcella
- 通讯作者:J. Bellazzini;Luigi Forcella
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