Concentration phenomena in nonlinear partial differential equations.

概括地说，我的项目领域是偏微分方程(PDE)。这是数学的一个分支，它使用微积分的工具来模拟自然界中的现象。事实上，物理学、生物学、经济学、社会学中的许多定律都可以用偏微分方程来表述。偏微分方程组的研究在数学中是一个非常广泛的领域，可以包括理论和更多的应用前景。例如，欧拉方程是一组描述运动流体的速度、压力和密度如何相关的偏微分方程组。这些方程忽略了Navier-Stokes方程中包含的粘性的影响。因此，欧拉方程的解只是真实流体模型的近似值。对于一些问题，如薄翼型在小迎角下的升力，欧拉方程的解提供了一个很好的现实模型。对于其他问题，如平板上边界层的增长，欧拉方程不能正确地模拟问题。我的主要兴趣是研究偏微分方程组的纯数学方面。在偏微分方程组的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？(如果没有，我们的模型就没有捕捉到一些本质的东西。)它们在初始数据的扰动下是稳定的吗？(如果没有，它们可能很难或不可能在自然界中观察到。)它们是否具有某种固有的对称性，以反映所模拟的潜在物理或生物现象？(自然界本质上是经济的，通常最简单的解决方案具有最大的对称性。)解在时间和空间上是平稳变化的，还是可以突然变化的(数学家称之为偏微分方程组中奇点的形成)？在这个建议中，我将针对特定的非线性偏微分方程组解决所有这些问题，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组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，人们观察到其解的奇性或某种形式的集中的形成。当解集中在较低维集上，或者依赖于解的某些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形式缺乏紧致性或解集中失去了正则性，这通常与所建模事件的相关情节有关。考虑化学反应引起的某些物质的爆炸或平面或桥梁上出现的断裂，我们提出了一些重要的非线性偏微分方程组的奇异性解的构造，例如不可压缩无粘流体的欧拉方程，超导中的Ginzburg-Landau模型，Sine-Gordon方程，趋化性中的Keller-Segel模型，以及给定的平均曲率问题。我的目标是阐述新的精致粘合技术来执行这些构造，并获得关于奇点为什么、在哪里和如何形成的准确描述。我的结果不仅对数学分析有意义，而且对几何流动、几何偏微分方程组和非线性偏微分方程边值问题也有意义。

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.