Singularity formations in non linear partial differential equations

在许多非线性发展的或平稳的偏微分方程组中，当模型的时间变量或参数接近极限值时，人们观察到其解的奇性或某种形式的集中的形成。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。本课题致力于研究一类非线性偏微分方程解的奇性形成问题。该项目打算回答的问题是：奇点会发生吗？触发奇点形成的机制是什么？奇点是在哪里(在空间上)以及何时(在时间上)形成的？这些奇点是什么形状的？奇点形成后会发生什么？新的数学方法论将在项目期间实施，包括首先确定可能奇点的形式和位置，并利用这一点构造一个良好的近似解。这第一步需要深入了解PDE所描述的模型。第二步是设计一种分析策略，以产生实际的解，而不是近似的解，例如通过使用微扰论点。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组，如欧拉方程、Navier-Stokes方程和湖泊方程，以及一些抛物型临界非线性偏微分方程组。最初的目的是以Kiselev和Sverak的论文《不可压缩二维Euler方程的解的小规模创建》为参考，构造具有有界初始涡度的解，以产生其梯度随时间以双指数形式增长的整体解。我们还计划研究当初始涡度高度集中在给定点附近时，在小粘性的Navier-Stokes二维模式中集中涡旋的演化。具体目标是使用粘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.