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主要基于偏微分方程。这些方程表达了流体背景下众所周知的物理定律,并描述了表征流动的量(例如速度和压力)如何随时间和空间变化。 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。 困难的原因之一来自于解决方案所表现出的高度不平凡的行为,其中包括复杂的小规模结构的出现和时间上的快速振荡。 这是方程中非线性的结果,它可以在不同尺度之间传递能量。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?飞机以什么速度会失速?关于方程的开放性理论问题也可以用相对简单的语言来表述:方程是否给出了流体演化的自洽描述,从某种意义上说,它们可以根据已知的当前状态唯一地预测流体的未来状态? 这是围绕方程的众所周知的开放数学问题之一,与解中奇点的可能发展密切相关。目前我们对方程的数学理解还不完整。方程的理论和实践方面的研究最终都有相同的目标:找到一些控制解的相对简单的参数集。人们希望通过确定正确的数量,能够对流动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围绕这些问题的几个开放数学问题。在更技术的层面上,提出的主题包括: 1.纳维-斯托克斯方程及相关方程的适定性、不适定性和唯一性,诸如:纳维-斯托克斯方程在自然能量空间中是否适定?根据最近有关尺度不变解决方案的工作,PI 预计这个问题的答案是否定的,但仍需要大量工作来证实这一点。 (这个问题也与具有有限能量初始日期的 Leray-Hopf 弱解的唯一性开放问题有关,但不一定是光滑的。)所提出的方法也应该适用于表面准地转方程,其中类似的问题是打开。欧拉方程还提出了许多有关适定性和稳定性的开放问题,其中一些问题将得到解决。即使在线性化水平上,许多问题也不清楚。从某种意义上说,这些比纳维-斯托克斯情况更微妙(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。2。复杂的 Ginzburg-Landau 方程的奇点和可能的非唯一性。该方程与纳维-斯托克斯方程具有相同的能量估计和相同的标度对称性。有强有力的证据表明,即使从平滑的初始条件开始,解决方案也可能会出现奇点。人们可以发展一种全局弱解理论,但这些理论是否能够唯一地预测系统的行为仍然是一个悬而未决的问题。唯一性问题对于评估方程的预测能力很重要。3.二维欧拉方程解的长期行为以及与该连接中使用的模型相关的偏微分方程问题。二维流的长期行为(例如与气象现象建模和气候预测相关)表现出一些显着的特征,但其数学理解仍然不完整。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出现的一些开放式偏微分方程问题,例如随机强迫流动中的不变测度的性质以及统计理论产生的稳态的性质。还将研究可以作为解决这些问题的良好模型的其他哈密顿偏微分方程。

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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非管井集水建筑物取水机理的物理模拟及计算模型研究
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