Variational Problems and Dynamics

变分问题和动力学

• 批准号: 1201354
• 负责人: Eric Carlen
• 金额: $ 45万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201354&HistoricalAwards=false
• 关键词: Variational; Problems; Dynamics

The proposed research program is focused on the interplay of dynamics and variational problems, with a special emphasis on quantitative stability estimates for sharp geometric inequalities. Such estimates and inequalities provide an effective means to derive properties of solutions of evolution equations and likewise, evolution equations can be used to derive variational inequalities. Exploiting this interplay has been very fruitful in the past work of the investigator, who plans to approach various problems in this framework. One is to prove new sharp non-local version of the Gagliardo-Nirenberg-Sobolev inequalities that arise in connection with the fast diffusion equation and porous medium equation. Such equations arise in the modeling of many phenomena, physical to biological, and the analytic problems to be investigated are chosen for their potential broader impact as well their intrinsic analytic interest. Another such problem concerns quantitiative stability for the Brun-Minkowski inequality and its application in the study of phase transitions. A similar philosophy applies as well to certain problems in kinetic theory, where the plan is to derive quantitative estimates on speed of approach to equilibrium for some inhomogeneous master equations of Kac type that have recently been used to model not only physical phenomena, but also phenomena in population biology.Many phenomena in science and technology can be modeled by evolution equations. One example that is the object of research in this proposal is the Keller Segel system, which models the aggregation, or the absence thereof, in the motion of bacteria. Understanding the behavior of solutions of these equations is both biologically and mathematically interesting. In particular, it is often observed that systems of many interacting agents (such as bacteria) or particles, either classical or quantum mechanical, evolve toward an equilibrium, and they do this at a certain speed. Determining the the speed of this process is vital to the understanding of many such models. This proposal concerns research on obtaining themathematical keys to such problems and their application in a broad range of fields. A central focus is solving certain maximization problems, and proving theorems asserting that any input that produces nearly maximal output must be close, in a an appropriate sense, to input that gives the exact maximum. As recent research of the investigator has shown, such theorems provide an effective means to unlock the information in models of the type described above, and the research proposed here will further this progress.

拟议的研究计划的重点是动力学和变分问题的相互作用，特别强调对尖锐的几何不等式的定量稳定性估计。这种估计和不等式提供了一种有效的手段来获得发展方程的解的性质，同样，发展方程可以用来获得变分不等式。调查员计划在这一框架内处理各种问题，在过去的工作中，利用这种相互作用取得了丰硕成果。一是证明了与快扩散方程和多孔介质方程有关的Gagliardo-Nirenberg-Sobolev不等式的新的非局部形式。这些方程出现在许多现象的建模中，从物理到生物，要研究的分析问题是根据其潜在的更广泛的影响以及其内在的分析兴趣来选择的。另一个这样的问题涉及Brun-Minkowski不等式的定量稳定性及其在相变研究中的应用。 类似的哲学也适用于动力学理论中的某些问题，其中计划是对一些Kac型的非齐次主方程的接近平衡的速度进行定量估计，这些主方程最近不仅被用来模拟物理现象，而且还被用来模拟种群生物学中的现象。本提案中研究对象的一个例子是Keller Segel系统，该系统对细菌运动中的聚集或不聚集进行建模。了解这些方程的解的行为在生物学和数学上都很有趣。 特别是，经常观察到许多相互作用的代理（如细菌）或粒子的系统，无论是经典的还是量子力学的，都朝着平衡发展，并且它们以一定的速度这样做。 确定这个过程的速度对于理解许多这样的模型至关重要。这一建议涉及研究如何获得这些问题的数学答案及其在广泛领域中的应用。一个中心焦点是解决某些最大化问题，并证明定理断言，任何输入，产生近最大输出必须接近，在适当的意义上，输入，给出确切的最大值。正如研究者最近的研究所示，这样的定理提供了一种有效的手段来解锁上述类型的模型中的信息，这里提出的研究将进一步推动这一进展。