Aspects of regularity theory for PDE
PDE 正则性理论的各个方面
基本信息
- 批准号:1101428
- 负责人:
- 金额:$ 41.2万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2011
- 资助国家:美国
- 起止时间:2011-06-01 至 2016-05-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The project addresses several open problems in the theory of partial differential equations. The problems arise primarily in the context of equations used in fluid mechanics (the Navier-Stokes equations and Euler's equations) or in the calculus of variations (regularity of minimizers of multi-dimensional variational integrals for vector-valued functions). For the Navier-Stokes equations the study will include the following areas of emphasis: (1) long-distance behavior of steady-state solutions; (2) regularity for special classes of solutions, such as the axi-symmetric solutions; and (3) regularity of related linear equations with low-regularity coefficients. In the case of Euler's equations, the project will focus on the following topics: (1) existence of periodic and quasi-periodic solutions; (2) the structure of the set of the steady-states of the two-dimensional equations; and (3) the relevance of various steady-states of the two-dimensional equations for two-dimensional statistical theories. The problems in the calculus of variations concern the stability of singularities. Unlike in the scalar case, the minimizers of regular variational functionals for vector-valued functions can have singularities. How stable are these singularities? This question will be addressed. The topic also has connections to nonlinear elasticity. Theoretical research in partial differential equations ultimately has a very practical goal, which is the understanding and prediction of behavior of solutions of the equations. For example, fluid flows are usually described by the Navier-Stokes equations. Do the solutions of these equations correctly describe what we see "in practice," and can we use the equations to make precise predictions about the flows? From what we know about these equations, the answers seem to be yes, but the mathematics of the equations is still not very well understood. Indeed, we do not yet have satisfactory mathematical explanations for the behavior of solutions that can be observed either in experiments or in computer simulations. The information we need to obtain from the computations can usually be formulated in simple and concrete terms. For instance, at what speed will an aircraft stall? The question is simple, whereas the behavior of the solutions underlying this question is complicated. Can we somehow manage the complexity by finding the most important mathematical parameters of the flow that are still "controlable"? When we study regularity of solutions, the situation is quite similar: there seems to be a very large spectrum of behaviors that solutions can exhibit. Can we obtain some control of the solutions by focusing on relatively few "right" parameters? What are the right parameters? This research project focuses on questions of this nature. If we can identify some good quantities that govern the behavior of the solutions and that we are able to control, then it becomes much easier to calculate the solutions themselves, since the theoretical information tells us where to focus our computational resources.
该项目解决了偏微分方程理论中的几个悬而未决的问题。这些问题主要出现在流体力学(纳维-斯托克斯方程和欧拉方程)或变分计算(向量值函数的多维变分积分的极小值的正则性)中使用的方程中。对于纳维-斯托克斯方程的研究将包括以下重点领域:(1)稳态解的长距离行为; (2)特殊类解的正则性,例如轴对称解; (3)相关低正则系数线性方程的正则性。就欧拉方程而言,该项目将重点关注以下主题:(1)周期和准周期解的存在性; (2) 二维方程组的稳态结构; (3)二维方程的各种稳态与二维统计理论的相关性。变分法中的问题涉及奇点的稳定性。与标量情况不同,向量值函数的正则变分泛函的极小值可以具有奇点。这些奇点有多稳定?这个问题将得到解答。该主题还与非线性弹性有关。偏微分方程的理论研究最终有一个非常实际的目标,那就是理解和预测方程解的行为。例如,流体流动通常由纳维-斯托克斯方程描述。这些方程的解是否正确地描述了我们“在实践中”看到的情况,我们可以使用这些方程对流量进行精确预测吗?从我们对这些方程的了解来看,答案似乎是肯定的,但方程的数学原理仍然不是很好理解。事实上,对于可以在实验或计算机模拟中观察到的解的行为,我们还没有令人满意的数学解释。我们需要从计算中获得的信息通常可以用简单而具体的术语来表述。例如,飞机以什么速度会失速?问题很简单,而该问题背后的解决方案的行为却很复杂。我们能否通过找到仍然“可控”的流中最重要的数学参数来管理复杂性?当我们研究解的规律性时,情况非常相似:解可以表现出非常大范围的行为。我们能否通过关注相对较少的“正确”参数来获得对解决方案的一些控制?什么是正确的参数?该研究项目重点关注这种性质的问题。如果我们能够确定一些控制解的行为并且我们能够控制的良好量,那么计算解本身就会变得容易得多,因为理论信息告诉我们将计算资源集中在哪里。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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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:
$ 41.2万 - 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:
$ 41.2万 - 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:
$ 41.2万 - 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:
$ 41.2万 - 项目类别:
Continuing Grant
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
- 批准号:
1362467 - 财政年份:2014
- 资助金额:
$ 41.2万 - 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:
$ 41.2万 - 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:
$ 41.2万 - 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:
$ 41.2万 - 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:
$ 41.2万 - 项目类别:
Standard Grant
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