Concentration phenomena in nonlinear partial differential equations.

非线性偏微分方程中的浓度现象。

• 批准号: EP/T008458/1
• 负责人: Monica Musso
• 金额: $ 38.43万
• 依托单位: University of Bath
• 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: https://gtr.ukri.org/projects?ref=EP%2FT008458%2F1
• 关键词: Concentration; phenomena; nonlinear; partial; differential

Broadly speaking, the area of my project is partial differential equations (PDEs). This is the branch of Mathematics which uses the tools of calculus to model phenomena in nature. Indeed, many laws in Physics, Biology, Economics, Social Studies, can be formulated as PDEs. The study of PDEs is a very broad field within Mathematics and can encompass both theoretical and more applied perspectives. For instance, Euler equations are a set of PDEs that describe how the velocity, pressure and density of a moving fluid are related. These equations neglect the effects of the viscosity which are included in the Navier-Stokes equations. A solution of the Euler equations is therefore only an approximation to a real fluids model. For some problems, like the lift of a thin airfoil at low angle of attack, a solution of the Euler equations provides a good model of reality. For other problems, like the growth of the boundary layer on a flat plate, the Euler equations do not properly model the problem.My main interest is for the purely mathematical aspects of the study of PDEs. The typical questions that arise in the study of PDEs include: Do solutions of a given equation (theoretically) exist? (If not, our model is not capturing something essential.) Are they stable under perturbations of the initial data? (If not, they may be difficult or impossible to observe in nature.) Do they have some inherent symmetry that reflects the underlying physical or biological phenomena being modeled? (Nature is intrinsically economical, and often the 'simplest' solutions have the most symmetry.) Do the solutions vary smoothly over time and space, or are abrupt changes possible (what mathematicians refer to as formation of singularities in PDEs)?In this proposal I will address all these questions for specific non-linear PDEs, with main emphasis on the last one: the mathematical analysis of formation of singularities. In many models, static or dynamic in nature, governed by non-linear PDEs, one observes the formation of singularities or some form of concentration of their solutions, as the time-variable or some parameter approaches a limit value. This happens when solutions become concentrated on lower-dimensional sets, or some expressions dependent on the solution become arbitrarily large. From a PDEs' point of view, this phenomenon reflects lack of compactness in the variational formulation of the problem or loss of regularity in the solution set, which is usually related with relevant episodes of the modeled event. Think of the explosion of some substance triggered by a chemical reaction or the appearance of fractures in planes or bridges.We propose the construction of solutions with singularities for some significant non-linear PDEs, such as for Euler equations for incompressible inviscid fluids, for Ginzburg-Landau model in superconductivity, for sine-Gordon equations, for Keller-Segel model in chemotaxis and for the prescribed mean curvature problem. My aim is to elaborate new refined gluing techniques to carry out these constructions and to derive precise descriptions on why, where and how formation of singularities takes place. My results will be of interest not only in Mathematical Analysis, but also in Geometric Flows, Geometric Partial Differential Equations and Boundary Value Problems for Nonlinear PDE's.

从广义上讲，我的项目领域是偏微分方程（PDE）。这是数学的一个分支，它使用微积分工具来模拟自然界的现象。事实上，物理学、生物学、经济学、社会研究中的许多定律都可以表述为偏微分方程。偏微分方程的研究是数学中一个非常广泛的领域，可以涵盖理论和更多应用的观点。例如，欧拉方程是一组偏微分方程，描述运动流体的速度、压力和密度之间的关系。这些方程忽略了纳维-斯托克斯方程中包含的粘度的影响。因此，欧拉方程的解只是真实流体模型的近似。对于某些问题，例如薄翼型在低攻角下的升力，欧拉方程的解提供了良好的现实模型。对于其他问题，例如平板上边界层的生长，欧拉方程无法正确模拟该问题。我的主要兴趣是偏微分方程研究的纯数学方面。偏微分方程研究中出现的典型问题包括：给定方程的解（理论上）是否存在？ （如果不是，我们的模型没有捕捉到一些重要的东西。）它们在初始数据的扰动下是否稳定？ （如果不是，它们在自然界中可能很难或不可能观察到。）它们是否具有一些固有的对称性，反映了正在建模的潜在物理或生物现象？ （大自然本质上是经济的，并且通常“最简单”的解具有最大的对称性。）解是否随时间和空间平滑变化，或者可能突然变化（数学家称之为偏微分方程中奇点的形成）？在本提案中，我将解决特定非线性偏微分方程的所有这些问题，主要强调最后一个：奇点形成的数学分析。在许多由非线性偏微分方程控制的静态或动态模型中，当时间变量或某些参数接近极限值时，人们会观察到奇点的形成或解的某种形式的集中。当解集中在低维集合上，或者某些依赖于解的表达式变得任意大时，就会发生这种情况。从偏微分方程的角度来看，这种现象反映了问题的变分公式缺乏紧凑性或解集失去了规律性，这通常与建模事件的相关事件有关。考虑由化学反应或平面或桥梁中出现裂缝引发的某些物质的爆炸。我们建议为一些重要的非线性偏微分方程构造具有奇点的解，例如不可压缩无粘流体的欧拉方程、超导性中的金兹堡-朗道模型、正弦-戈登方程、趋化性中的凯勒-塞格尔模型以及 规定的平均曲率问题。我的目标是详细阐述新的精制粘合技术来执行这些构造，并得出关于奇点形成的原因、地点和方式的精确描述。我的结果不仅对数学分析感兴趣，而且对几何流、几何偏微分方程和非线性偏微分方程的边值问题也很感兴趣。

项目成果

Leapfrogging vortex rings for the 3-dimensional incompressible Euler equations

3 维不可压缩欧拉方程的蛙跳涡环

• DOI: 10.48550/arxiv.2207.03263
• 发表时间: 2022
• 作者: Davila J

Geometry driven type II higher dimensional blow-up for the critical heat equation

• DOI: 10.1016/j.jfa.2020.108788
• 发表时间: 2017-10
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Manuel del Pino;M. Musso;Juncheng Wei

Infinite-time blow-up for the 3-dimensional energy-critical heat equation

• DOI: 10.2140/apde.2020.13.215
• 发表时间: 2017-05
• 期刊: Analysis & PDE
• 影响因子: 2.2
• 作者: M. Pino;M. Musso;Juncheng Wei

Doubling the equatorial for the prescribed scalar curvatureproblem on ${\mathbb{S}}^N$

将 ${mathbb{S}}^N$ 上规定的标量曲率问题的赤道线加倍

• DOI: 10.21203/rs.3.rs-2470846/v1
• 发表时间: 2023
• 作者: Duan L

High energy sign-changing solutions for Coron's problem

科隆问题的高能变号解决方案

• DOI: 10.1016/j.jde.2020.09.021
• 发表时间: 2021
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Deng S