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

JabinDMS-1614537 对流或传输机制是物理学和生物科学中几种现象的关键特征。 这个项目的重点是在可压缩流体力学和系统的许多“粒子”的机制。可压缩流体力学包括大量不同环境下的模型：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（如游泳细菌）或“外来”例子，如太阳事件或光子辐射。 粒子系统通常涉及大量的耦合方程，每个粒子一个。 粒子在这里可以代表许多不同的对象：在物理学中，它们可以代表等离子体中的离子和电子，或流体中的分子，甚至在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，粒子是个体的“代理人”。“不稳定性可以在所有这些系统中发展，并可能表现为流体质量密度的振荡或粒子之间的碰撞（或接近碰撞）。 更好地理解这种行为可能会产生重要的后果。 例如，对微生物群集运动的深入了解将有助于开发新的生物技术。 降低这类系统复杂性的主要挑战是理解对流或传输（流体或粒子的质量）如何以及多快或多慢地放大这种振荡。 研究生也参与了该项目的工作。 该项目为研究多粒子系统和非线性对流方程的不稳定性提供了一个新的数学框架。 它统一的方法来控制粒子之间的碰撞，所需的奇异力，和那些传播规律性的对流方程，需要限制非线性项的振荡。 新的方法依赖于一个直接控制的正则性的解决方案，通过加倍的变量，需要微妙和明确的换向器估计。 显然非常不同的问题之间的这种自然相互作用导致了新的见解，并打破了一些更现实的模型的障碍：相互作用的粒子系统与物理上现实的和无界的力量或随机碰撞的速度，导致退化扩散;流体力学模型与不稳定的状态法律或各向异性条款。 研究生也参与了该项目的工作。

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.