Regularization Properties of Nonstandard Diffusions

随着时间的推移，扩散过程使空间分布变得更加规则。有许多系统表现出这种行为。整个物体的温度值、某种化学物质的浓度或迁移中的人口密度都是表现扩散行为的量的一些例子。在数学分析中，扩散方程的主要例子是热方程，或者更一般的抛物型偏微分方程。本研究项目的目的是研究非标准扩散。该项目探索在抛物方程的经典框架之外的非线性方程的正则化效应。该研究研究了非局部方程，其中任意点的值的演变取决于全局图像，并将这些方法应用于稀气体中粒子密度的统计力学建模方程中。该项目还研究了守恒定律方程，一类基本的偏微分方程，并旨在阐明正则化性质。该项目将有研究生和博士后参与研究。研究结果将通过出版物、会议和暑期学校传播。非局部方程的研究，更确切地说是积分微分方程，是一个非常活跃的研究领域。最近的工作表明，该领域发展起来的一些思想和方法可以用于从统计力学中获得玻尔兹曼方程的正则性估计。本研究项目旨在证明如果玻尔兹曼方程的解出现奇点，则一定有水动力量是无界的。这些流体动力学量是质量、能量和熵密度，它们对应于流体的物理有意义的性质。因此，该项目旨在确定，如果在由玻尔兹曼方程给出的流体的介观描述中存在奇点，那么它必须在宏观上是明显的。玻尔兹曼方程可以改写为抛物型积分微分方程，其流体动力量的限制决定了相应核的非简并性。积分微分方程和玻尔兹曼方程之间的这种联系对于前一学科的进一步发展也是有用的指导。将抛物方程的方法应用于守恒定律方程的正则化机制的研究可以说是更令人惊讶的。在标量守恒定律中，解的传输速度取决于每一点的值。对于光滑初始数据，激波奇点是不可避免的；然而，当从非常粗糙的初始数据开始时，相同的冲击机制迫使一些自组织转化为正则化效应。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The process of diffusion makes a distribution in space become more regular as time goes by. There are many systems that exhibit this behavior. The values of the temperature throughout an object, the concentration of some chemical, or the population density in migration are some examples of quantities that show diffusive behavior. In mathematical analysis, the prime example of a diffusive equation is the heat equation, or the more general family of parabolic partial differential equations. The purpose of this research project is to study non-standard diffusion. The project explores regularization effects in nonlinear equations that are outside the classical framework of parabolic equations. The research studies non-local equations, where the evolution of the value at any point depends on the global picture, and applies those methods in equations from statistical mechanics modeling the density of particles in a dilute gas. The project also studies conservation law equations, a fundamental class of partial differential equation, and aims to explicate regularization properties there as well. The project will involve graduate students and postdoctoral associates in the research. Results will be disseminated through publications, conferences, and summer schools.The study of non-local equations, and more precisely integro-differential equations, is a very active area of research. Recent work indicates that some of the ideas and methods developed in that area can be used to obtain regularity estimates for the Boltzmann equation from statistical mechanics. This research project aims at proving that if the solution of the Boltzmann equation ever develops a singularity, then some hydrodynamic quantity must be unbounded. These hydrodynamic quantities are the mass, energy, and entropy density, which correspond to physically meaningful properties of the fluid. Thus, the project aims to establish that if there is a singularity in the mesoscopic description of the fluid given by the Boltzmann equation, it must be apparent macroscopically. The Boltzmann equation can be rewritten as a parabolic integro-differential equation, and the restriction on its hydrodynamic quantities determines the non-degeneracy of the corresponding kernel. This connection between integro-differential equations and the Boltzmann equation is also useful as a guide for further development of the former subject. The study of regularization mechanisms using methods from parabolic equations applied to conservation law equations is arguably even more surprising. In scalar conservation laws, the solution is transported following a velocity that depends on the value at every point. Shock singularities are unavoidable for smooth initial data; however, the same shock mechanism forces some self-organization that translates into a regularization effect when starting from very rough initial data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.