Variational Problems and Dynamics

拟议的研究计划集中于动力学和变分问题的相互作用，特别强调尖锐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方法来推导发展方程的解的性质，同样，发展方程也可以用来推导变分不等式。在调查员过去的工作中，利用这种相互作用非常富有成效，他计划在这个框架内处理各种问题。其一是证明了与快速扩散方程和多孔介质方程有关的新的非局部形式的Gagliardo-Nirenberg-Sobolev不等式。这样的方程出现在许多现象的建模中，从物理到生物，选择要研究的分析问题是因为它们潜在的更广泛的影响以及内在的分析兴趣。另一个这样的问题涉及Brun-Minkowski不等式的量化稳定性及其在相变研究中的应用。类似的哲学也适用于运动论中的某些问题，其中计划对一些Kac型非齐次主方程的接近平衡的速度进行定量估计，这些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。科学和技术中的许多现象都可以用演化方程来模拟。作为本提案研究对象的一个例子是Keller Segel系统，该系统模拟细菌运动中的聚集或不聚集。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，人们经常观察到许多相互作用剂(如细菌)或粒子的系统，无论是经典的还是量子力学的，都会进化到一种平衡，并且它们以一定的速度做到这一点。确定这一过程的速度对于理解许多这样的模型至关重要。这项建议涉及获得此类问题的数学关键字及其在广泛领域中的应用的研究。一个中心焦点是解决某些最大化问题，并证明定理，这些定理断言任何产生几乎最大输出的输入在适当的意义上都必须接近给出精确最大值的输入。正如研究者最近的研究表明，这些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，而这里提出的研究将进一步推动这一进展。

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.