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

非线性偏微分方程的完备性和长时间行为

基本信息

批准号：
1362467
负责人：
Vladimir Sverak
金额：
$ 32.4万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-07-01 至 2017-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362467&HistoricalAwards=false
关键词：
Aspects well possedeness long time

项目摘要

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.

流体流的数学描述主要基于部分微分方程。这些方程式在流体的背景下表达了众所周知的物理定律，并描述了表征流量的数量，例如速度和压力会在时空发生变化。事实证明，即使在大型计算机的帮助下，方程式也很难解决。造成困难的原因之一来自解决方案表现出的高度非平凡的行为，其中包括复杂的小规模结构的出现和及时的快速振荡。这是方程中非线性的结果，该方程式可以在各种尺度之间传递能量。流体运动可能很复杂，但是实际问题在某种意义上很简单：龙卷风会形成吗？飞机会以哪个速度停下来？关于方程式的开放理论问题也可以用相对简单的语言提出：方程是否给出了对流体演化的自洽描述，因为它们可以基于已知的当前状态独特地预测流体的未来状态？这是围绕方程式的众所周知的开放数学问题之一，与解决方案中奇异性的可能发展密切相关。目前，我们对方程式的数学理解是不完整的。方程式的理论和实际方面的研究最终都具有相同的目标：找到一些控制解决方案的相对简单的参数集。希望通过确定正确的数量，可以对流量进行良好的描述，并充分表征其重要特征。该研究项目的主要工作是针对这些问题的几个开放数学问题。在更技术层面上，拟议的主题包括：1。 Navier-Stokes方程和相关方程式的适应性，不足和独特性，诸如：Navier-Stokes方程在自然能量空间中是否适合？根据有关规模不变解决方案的最新工作，PI期望该问题的答案是负面的，但是仍然需要进行大量工作来确认这一点。（这个问题还与有限能量的初始日期但不一定光滑的Leray-Hopf弱解决方案的唯一性问题有关。）所提出的方法也应适用于表面准蜜饯方程，在此开放类似的问题。 Euler方程还提出了许多有关适应性和稳定性的开放问题，其中一些将被解决。即使在线性化水平上，许多问题也不清楚。这些在某种意义上比在Navier-Stokes案例中更微妙（由于连续光谱的强大作用），但应提供对低粘度流量的有价值的见解。2。复杂的金茨堡 - 兰道方程的奇异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值和相同的缩放对称性。有强有力的证据表明，即使从光滑的初始条件开始，解决方案也可以发展出奇异性。一个人可以发展出全球弱解决方案的理论，但是如果这些独特地预测系统的行为，它仍然是开放的。唯一性问题对于评估方程的预测能力很重要3。与该连接中使用的模型相关的2D Euler方程和PDE问题的解决方案的长期行为。 2D流的长期行为（例如，对于气象现象的建模和对气候的预测，相关）表现出一些惊人的特征，这些特征的数学理解仍然不完整。与统计力学和其他无限二维Hamiltonian PDE有牢固的联系。这项研究将解决在这种情况下引起的一些开放性PDE问题，例如流动中不变措施的特性，具有随机强迫和由统计理论引起的稳态的特性。还将研究其他可以作为这些问题的良好模型的汉密尔顿PDE。