Singularity formations in non linear partial differential equations

非线性偏微分方程中的奇异性形成

• 批准号: 2278691
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2278691
• 关键词: Singularity; formations; non; linear; partial

In many non-linear evolutionary or stationary partial differential equations (PDEs), one observes the formation of singularities or some form of concentration of their solutions, as the time-variable or a parameter of the model approaches a limit value. In that case solutions become highly concentrated on lower-dimensional sets, thus losing smoothness and approaching a singular limit. This project is devoted to the study of formation of singularities for solutions of classes of non-linear PDEs. The questions the project intend to answer are: Do singularities occur? What is the mechanism that triggers the formation of singularities? Where (in space) and when (in time) do singularities develop? What is the shape of such singularities? What happens after the formation of singularities? The novel mathematical methodology that will be carried out during the project consists of identifying first the form and location of a possible singularity and using this to construct a good approximate solution. This first step requires a deep understanding of the model the PDEs are describing. The second step consists of designing an analytic strategy to produce an actual solution rather than an approximate one, for instance by using a perturbation argument. The student will consider some model PDEs which describe the motion of an incompressible fluid in dimension two, such as Euler equations, Navier-Stokes equations and the lake equations, as well as some parabolic critical non-linear PDEs. The initial aim is to construct solutions with bounded initial vorticity which produce a global solution whose gradient grows in time as a double exponential for the lake equation, using as a reference the paper 'Small scale creation for solutions of the incompressible two dimensional Euler equation' by Kiselev and Sverak. The plan is also to investigate the evolution of concentrated vorticities in the Navier-Stokes 2-dimensional model for small viscosity, when the initial vorticity is highly concentrated around a given number of points. The specific aim is to build such solutions using gluing techniques.

在许多非线性进化或固定部分偏微分方程（PDE）中，人们观察到奇点或某种形式的溶液浓度，因为该模型的时间变化或参数接近限制值。在这种情况下，解决方案高度集中在较低维度的集合上，从而失去平滑度并接近奇异极限。该项目致力于研究非线性PDE类别解决方案的奇异性的形成。项目打算回答的问题是：奇点会发生吗？触发奇异性形成的机制是什么？ （在太空中）的何处以及何时（时间）何时发展？这种奇异性的形状是什么？形成奇点后会发生什么？在项目期间将进行的新型数学方法包括首先识别可能的奇异性的形式和位置，并使用它来构建良好的近似解决方案。第一步需要对PDE所描述的模型有深入的了解。第二步包括设计一种分析策略来产生实际解决方案而不是大约解决方案，例如使用扰动参数。学生将考虑一些模型PDE，这些模型描述了二维尺寸的不可压缩流体的运动，例如Euler方程，Navier-Stokes方程和湖泊方程，以及某些抛物线临界非线性PDES。最初的目的是构建具有有限的初始涡度的解决方案，该解决方案产生了一个全局溶液，其梯度随着湖泊方程的双重指数而生长，作为参考纸的“小规模创建”，用于kiselev和sverak的不可压缩的二维Euler方程。该计划还旨在调查小粘度的Navier-Stokes二维模型中浓缩涡度的演变，而初始涡度高度集中在给定数量的点围绕数量的点。具体的目的是使用胶合技术构建此类解决方案。