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平稳偏微分方程（PDEs）中，当模型的时间变量或参数接近一个极限值时，人们观察到奇点的形成或其解的某种形式的集中。在这种情况下，解变得高度集中在低维集合上，从而失去平滑性并接近奇异极限。本课题主要研究一类非线性偏微分方程解奇点的形成。这个项目想要回答的问题是：奇点会发生吗？触发奇点形成的机制是什么？奇点在哪里（在空间上）和何时（在时间上）发展？奇点的形状是什么？奇点形成后会发生什么？新数学方法将在项目中实施，包括首先确定可能的奇点的形式和位置，并使用它来构建一个很好的近似解。这第一步需要对pde所描述的模型有深刻的理解。第二步包括设计一个解析策略来产生一个实际的解决方案，而不是一个近似的解决方案，例如通过使用摄动参数。学生将考虑一些描述二维不可压缩流体运动的模型偏微分方程，如欧拉方程、纳维-斯托克斯方程和湖方程，以及一些抛物线临界非线性偏微分方程。最初的目的是构造具有有界初始涡度的解，该解产生梯度随时间增长的湖方程全局解，作为双指数，参考Kiselev和Sverak的论文“不可压缩二维欧拉方程解的小尺度创建”。该计划还将研究小粘度下Navier-Stokes二维模型中集中涡度的演变，当初始涡度高度集中在给定数量的点周围时。具体目标是使用粘合技术构建这样的解决方案。