Aspects of well-possedeness and long time behavior for non-linear PDEs

流体流动的数学描述大多基于偏微分方程组。这些方程表达了流体中众所周知的物理定律，并描述了表征流动的量，如速度和压力将如何在时间和空间中变化。事实证明，即使在大型计算机的帮助下，这些方程也很难求解。困难的一个原因来自于解表现出的高度非平凡的行为，包括复杂的小尺度结构的出现和快速的时间振荡。这是由于方程中的非线性，可以在不同的尺度之间传递能量。流体运动可能很复杂，但实际问题在某种意义上很简单：龙卷风会形成吗？飞机会以多快的速度失速？关于这些方程的开放的理论问题也可以用一种相对简单的语言来表述：这些方程是否给出了对流体演化的自洽描述，从某种意义上说，它们可以基于已知的当前状态唯一地预测流体的未来状态？这是围绕方程的著名的公开数学问题之一，与解中可能出现的奇异性密切相关。我们对这些方程的数学理解目前还不完整。对该方程的理论和实践方面的研究最终都有一个相同的目标：找到一些相对简单的参数集来控制解。人们希望，通过确定正确的数量，人们将能够很好地描述流动并充分描述其重要特征。本课题的主要工作就是围绕这些问题展开研究，在更高的技术层面上，本文的研究内容包括：1.研究了Navier-Stokes方程及其相关方程的适定性、不适定性和唯一性，如：在自然能量空间中，Navier-Stokes方程是否适定？基于最近关于尺度不变解的工作，PI预计这个问题的答案是否定的，但仍需要大量的工作来证实这一点。(这个问题也与Leray-Hopf弱解的唯一性公开问题有关，该弱解的初始日期为有限能量，但不一定是光滑的。)所提出的方法也适用于地表准地转方程，其中类似的问题尚待解决。欧拉方程还提出了一些关于适定性和稳定性的公开问题，其中一些将被解决。即使在线性化的层面上，许多问题也是不清楚的。从某种意义上讲，这比纳维-斯托克斯的情况更微妙(由于连续谱的强大作用)，但应该能为低粘度流动提供有价值的见解。复Ginzburg-Landau方程的奇性和可能的非唯一性。该方程具有与Navier-Stokes方程相同的能量估计和标度对称性。有强有力的证据表明，即使从光滑的初始条件开始，解也会产生奇点。人们可以发展出一种全球弱解的理论，但这些解是否能唯一地预测系统的行为仍是一个未知数。唯一性问题对于评估方程的预测能力是很重要的。二维欧拉方程的解的长时间行为和与此相关的模型相关的偏微分方程解。二维流的长期行为(例如与气象现象的模拟和对气候的预测有关)显示出一些突出的特征，其数学理解仍然不完整。这与统计力学和其他无限维哈密顿偏微分方程有很强的联系。这项研究将解决在这一背景下出现的一些公开的偏微分方程问题，如随机强迫流中不变测度的性质和统计理论中产生的稳态性质。其他可以作为这些问题的良好模型的哈密顿偏微分方程组也将被研究。

Mathematical description of fluid flows is mostly based on partial differential equations. These equations express well-known physical laws in the context of fluids and describe how quantities characterizing the flow, such as the velocity and the pressure will change in time and space. It turns out the equations are difficult to solve, even with the help of large computers. One reason for the difficulties comes from the highly non-trivial behavior exhibited by the solutions, which includes the emergence of complicated small-scale structures and fast oscillations in time. This is a result of the non-linearity in the equations, which can transfer energy between various scales. The fluid motion can be complicated, but the practical questions are in some sense simple: will a tornado form? At which speed will a plane stall? Open theoretical questions about the equations can also be formulated in a relatively simple language: do the equations give a self-consistent description of the fluid evolution, in the sense that they can uniquely predict the future state of the fluid based on a known current state? This is one of the well-known open mathematical problems surrounding the equations, closely related to the possible development of singularities in the solutions. Our mathematical understanding of the equations is currently incomplete. The research at both theoretical and practical aspects of the equation has ultimately the same goal: to find some relatively simple set of parameters which control the solutions. One hopes that by identifying the right quantities, one will be able to give a good description of the flow and sufficiently characterize its important features. The main effort of this research project is aimed at several open mathematical problems surrounding these issues.At a more technical level, the proposed topics include:1. Well-posedness, ill-posedness and uniqueness for the Navier-Stokes equation and related equations, questions such as: Is the Navier-Stokes equation well-posed in the natural energy space? Based on recent work concerning scale-invariant solutions, the PI expects that the answer to this questions is negative, but significant work is still needed to confirm this. (The question is also related to the open problem of uniqueness of the Leray-Hopf weak solutions with initial date of finite energy, but not necessarily smooth.) The proposed methods should also work for the surface quasi-geostrophic equation, where similar questions are open. The Euler equation also presents a number of open problem concerning well-posedness and stability, some of which will be addressed. Many issues are not clear even at the linearized level. These are in some sense more subtle than in the Navier-Stokes case (due to the strong role of continuous spectra), but should provide valuable insights into low-viscosity flows.2. Singularities and possible non-uniqueness for the complex Ginzburg-Landau equation. This equation has the same energy estimates and the same scaling symmetry as the Navier-Stokes equation. There is strong evidence that solution can develop singularities even when starting from smooth initial conditions. One can develop a theory of global weak solutions, but it remains open whether these uniquely predict the behavior of the system. The question of uniqueness is important for assessing the predictive power of the equation.3. Long-time behavior of solutions for the 2d Euler equation and PDE problems associated with models used in that connection. The long-time behavior of 2d flows (relevant for example for modelling of meteorological phenomena and making predictions concerning climate) exhibits some striking features whose mathematical understanding remains incomplete. There are strong connections to Statistical Mechanics and other infinite-dimensional Hamiltonian PDEs. The research will address some of the open PDE problems arising in this context, such as the properties of invariant measures in flows with stochastic forcing and properties of steady-states arising from statistical theories. Other Hamiltonian PDEs which can serve as good models for these questions will also be studied.