Singularity formations in non linear partial differential equations

在许多非线性演化或定常偏微分方程（PDE）中，当模型的时间变量或参数接近极限值时，可以观察到奇点的形成或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，从而失去光滑性并接近奇异极限。本计画致力于研究非线性偏微分方程类解的奇性形成。该项目打算回答的问题是：奇点是否发生？触发奇点形成的机制是什么？在哪里（在空间中）和何时（在时间中）奇点发展？这种奇点的形状是什么？奇点形成后会发生什么？该项目将采用新的数学方法，首先确定可能的奇点的形式和位置，并利用它来构造一个良好的近似解。第一步需要对偏微分方程描述的模型有深入的理解。第二步是设计一个分析策略，以产生一个实际的解决方案，而不是一个近似的，例如通过使用扰动参数。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，Navier-Stokes方程和湖泊方程，以及一些抛物临界非线性偏微分方程。最初的目标是构建有界初始涡度的解决方案，产生一个整体的解决方案，其梯度增长的时间作为一个双指数的湖方程，使用作为参考文件“小规模创建解决方案的不可压缩的二维欧拉方程”由基谢廖夫和Sverak。该计划还将调查小粘度纳维尔-斯托克斯二维模型中的集中涡度的演变，当初始涡度高度集中在给定数量的点周围时。具体目标是使用胶合技术构建这样的解决方案。

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.