A novel paradigm for nonlinear convection models and large systems of particles

非线性对流模型和大型粒子系统的新范例

基本信息

批准号：
1614537
负责人：
Pierre-Emmanuel Jabin
金额：
$ 40万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1614537&HistoricalAwards=false
关键词：
novel paradigm nonlinear convection models

项目摘要

JabinDMS-1614537 Convection or transport mechanisms are a critical feature of several phenomena in physics and the biosciences. This project focuses on such mechanisms in compressible fluid mechanics and systems of many "particles." Compressible fluid mechanics includes a large set of models in very diverse settings: geophysical fluids with gravity in large scale fluids (Earth atmosphere), biological fluids (such as swimming bacteria), or "exotic" examples such as solar events or photon radiation. Systems of particles typically involve a very large number of coupled equations, one for each particle. Particles can here represent many different objects: In physics they can represent ions and electrons in plasmas, or molecules in a fluid, or even galaxies in some cosmological models; in the biosciences particles typically model micro-organisms (cells or bacteria); in economics or social sciences, particles are individual "agents." Instabilities can develop in all those systems and may manifest as oscillations in the mass density of the fluid or as collisions (or near collisions) between particles. A better understanding of such behavior can have important consequences. For example, insights on the swarming motility of micro-organisms would help the development of new biotechnologies. The main challenge to reduce the complexity of such systems is to understand how, and how quickly or slowly, the convection or transport (of mass in the fluid or of the particles) can amplify such oscillations. Graduate students are included in the work of the project. The project offers a new mathematical framework to study instabilities in many-particles systems and nonlinear convection equations. It unifies the methods to control collisions between particles, required by singular forces, and the ones to propagate regularity in convection equations, required to limit oscillations in nonlinear terms. The new method relies on a direct control of the regularity of the solution through a doubling of variables and requires delicate and explicit commutator estimates. This natural interplay between apparently very different sets of problems leads to new insights and breaks some of the barriers to more realistic models: Systems of interacting particles with physically realistic and unbounded forces or random collisions in velocity, leading to degenerate diffusion; models of fluid mechanics with thermodynamically unstable state laws or anisotropic terms. Graduate students are included in the work of the project.

JabinDMS-1614537 对流或传输机制是物理学和生物科学中多种现象的关键特征。该项目重点研究可压缩流体力学和许多“粒子”系统中的此类机制。可压缩流体力学包括在非常不同的环境中的大量模型：大规模流体（地球大气）中具有重力的地球物理流体、生物流体（例如游动的细菌）或“奇异”示例，例如太阳事件或光子辐射。粒子系统通常涉及大量耦合方程，每个粒子一个。粒子在这里可以代表许多不同的物体：在物理学中，它们可以代表等离子体中的离子和电子，或者流体中的分子，甚至某些宇宙学模型中的星系；在生物科学中，粒子通常模拟微生物（细胞或细菌）；在经济学或社会科学中，粒子是个体“主体”。所有这些系统中都会出现不稳定性，并可能表现为流体质量密度的振荡或颗粒之间的碰撞（或接近碰撞）。更好地理解此类行为可能会产生重要的后果。例如，对微生物集群运动的洞察将有助于新生物技术的发展。降低此类系统复杂性的主要挑战是了解（流体或颗粒中的质量）对流或传输如何以及多快或多慢地放大此类振荡。研究生参与该项目的工作。该项目提供了一个新的数学框架来研究多粒子系统和非线性对流方程的不稳定性。它统一了控制奇异力所需的粒子之间碰撞的方法，以及限制非线性项中的振荡所需的对流方程中传播规律性的方法。新方法依赖于通过变量加倍来直接控制解的规律性，并且需要精细且明确的换向器估计。明显不同的问题组之间的这种自然相互作用带来了新的见解，并打破了更现实模型的一些障碍：具有物理现实和无限力或速度随机碰撞的粒子相互作用系统，导致简并扩散；具有热力学不稳定状态定律或各向异性项的流体力学模型。研究生参与该项目的工作。