Variational Problems and Dynamics

提出的研究计划侧重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。这种估计和不等式为推导演化方程解的性质提供了一种有效的手段，同样，演化方程也可以用来推导变分不等式。利用这种相互作用在研究者过去的工作中非常富有成效，他们计划在这个框架中处理各种问题。一是证明与快速扩散方程和多孔介质方程有关的Gagliardo-Nirenberg-Sobolev不等式的新的尖锐非局部版本。这样的方程出现在许多现象的建模中，从物理到生物，要研究的分析问题被选择为它们潜在的更广泛的影响以及它们内在的分析兴趣。另一个这样的问题涉及布朗-闵可夫斯基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运动论中的某些问题，其中的计划是对一些非齐次的Kac型主方程接近平衡的速度进行定量估计，这些方程最近不仅用于模拟物理现象，而且还用于模拟种群生物学中的现象。科学技术中的许多现象都可以用进化方程来模拟。本提案的研究对象之一是Keller Segel系统，该系统模拟了细菌运动中的聚集或缺乏聚集。理解这些方程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媒介（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。确定这一过程的速度对于理解许多此类模型至关重要。这一建议是关于获得这些问题的数学关键及其在广泛领域中的应用的研究。中心焦点是解决某些最大化问题，并证明断言任何产生接近最大输出的输入必须在适当的意义上接近产生确切最大值的输入的定理。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开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.