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:
  • 金额:
    $ 32.4万
  • 依托单位:
  • 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

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.
流体流动的数学描述大多基于偏微分方程。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,如速度和压力如何随时间和空间变化。事实证明,即使有大型计算机的帮助,这些方程也很难解。困难的一个原因来自解所表现出的高度非平凡的行为,其中包括出现复杂的小尺度结构和时间上的快速振荡。这是方程非线性的结果,它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?飞机在什么速度下会失速?关于这些方程的开放性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流体演化的自洽描述,在某种意义上,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?这是围绕方程的一个众所周知的开放数学问题,与解中奇点的可能发展密切相关。我们对这些方程的数学理解目前还不完整。方程的理论和实践研究最终都有一个共同的目标: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,并充分表征其重要特征。这个研究项目的主要工作是针对围绕这些问题的几个开放的数学问题。在更技术性的层面上,建议的主题包括:Navier-Stokes方程及相关方程的适定性,病态性和唯一性,如:Navier-Stokes方程在自然能量空间中适定性吗?根据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。(该问题也与初始日期为有限能量的Leray-Hopf弱解的唯一性问题有关,但不一定是光滑的。)所提出的方法也应该适用于曲面准地转方程,在那里类似的问题是开放的。欧拉方程还提出了许多关于适定性和稳定性的开放性问题,其中一些问题将在本文中讨论。即使在线性化的水平上,许多问题也不清楚。在某种意义上,这些比在Navier-Stokes情况下更微妙(由于连续光谱的强大作用),但应该为低粘度流动提供有价值的见解。复金兹堡-朗道方程的奇异性和可能的非唯一性。这个方程和Navier-Stokes方程有相同的能量估计和相同的尺度对称。有强有力的证据表明,即使从光滑初始条件出发,解也可以发展出奇点。人们可以发展出一种全局弱解理论,但这些理论是否能唯一地预测系统的行为,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。二维欧拉方程解的长期行为和与该连接中使用的模型相关的PDE问题。二维流动的长期行为(例如与气象现象的建模和有关气候的预测有关)显示出一些显著的特征,其数学理解仍然不完整。它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。该研究将解决在此背景下产生的一些开放的偏微分方程问题,例如随机强迫流动中不变测度的性质和统计理论产生的稳态性质。其他可以作为这些问题的良好模型的哈密顿偏微分方程也将被研究。

项目成果

期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Dynamics of geodesic flows with random forcing on Lie groups with left-invariant metrics
具有左不变度量的李群上具有随机力的测地流动力学
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Vladimir Sverak其他文献

Vladimir Sverak的其他文献

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{{ truncateString('Vladimir Sverak', 18)}}的其他基金

Topics in the Analysis of Nonlinear Partial Differential Equations
非线性偏微分方程分析专题
  • 批准号:
    2247027
  • 财政年份:
    2023
  • 资助金额:
    $ 32.4万
  • 项目类别:
    Standard Grant
Regularity, Stability, and Uniqueness Questions for Certain Non-Linear Partial Differential Equations
某些非线性偏微分方程的正则性、稳定性和唯一性问题
  • 批准号:
    1956092
  • 财政年份:
    2020
  • 资助金额:
    $ 32.4万
  • 项目类别:
    Standard Grant
The Twentieth Riviere-Fabes Symposium
第二十届Riviere-Fabes研讨会
  • 批准号:
    1665006
  • 财政年份:
    2017
  • 资助金额:
    $ 32.4万
  • 项目类别:
    Standard Grant
Questions in Nonlinear Partial Differential Equations
非线性偏微分方程问题
  • 批准号:
    1664297
  • 财政年份:
    2017
  • 资助金额:
    $ 32.4万
  • 项目类别:
    Continuing Grant
The Sixteenth Riviere-Fabes Symposium
第十六届Riviere-Fabes研讨会
  • 批准号:
    1304998
  • 财政年份:
    2013
  • 资助金额:
    $ 32.4万
  • 项目类别:
    Standard Grant
FRG: Collaborative Research: Singularities, mixing and long time behavior in nonlinear evolution
FRG:协作研究:非线性演化中的奇异性、混合和长期行为
  • 批准号:
    1159376
  • 财政年份:
    2012
  • 资助金额:
    $ 32.4万
  • 项目类别:
    Standard Grant
Aspects of regularity theory for PDE
PDE 正则性理论的各个方面
  • 批准号:
    1101428
  • 财政年份:
    2011
  • 资助金额:
    $ 32.4万
  • 项目类别:
    Continuing Grant
Riviere-Fabes Symposium
里维埃-法贝斯研讨会
  • 批准号:
    1004156
  • 财政年份:
    2010
  • 资助金额:
    $ 32.4万
  • 项目类别:
    Standard Grant
Problems in Nonlinear Partial Differential Equations
非线性偏微分方程问题
  • 批准号:
    0800908
  • 财政年份:
    2008
  • 资助金额:
    $ 32.4万
  • 项目类别:
    Continuing Grant
Ninth Riviere-Fabes Symposium on Analysis and PDE, April 2006
第九届 Riviere-Fabes 分析和偏微分方程研讨会,2006 年 4 月
  • 批准号:
    0606843
  • 财政年份:
    2006
  • 资助金额:
    $ 32.4万
  • 项目类别:
    Standard Grant

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