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个现象的关键特征。这个项目的重点是可压缩流体力学和许多“粒子”系统中的这种机制。可压缩流体力学包括在非常不同的环境中的一大组模型：大尺度流体(地球大气)中具有重力的地球物理流体、生物流体(如游动的细菌)或诸如太阳事件或光子辐射的“奇异”例子。粒子系统通常涉及大量的耦合方程，每个粒子对应一个方程。在这里，粒子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，甚至在一些宇宙模型中，甚至可以代表星系；在生物科学中，粒子通常代表微生物(细胞或细菌)；在经济学或社会科学中，粒子是单独的“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，对微生物成群运动的洞察将有助于新生物技术的发展。降低这类系统的复杂性的主要挑战是了解对流或输送(流体或颗粒中的质量)如何以及以多快或多慢的速度放大这种振荡。研究生被包括在该项目的工作中。该项目为研究多粒子系统和非线性对流方程中的不稳定性提供了一个新的数学框架。它统一了奇异力所要求的控制粒子间碰撞的方法，以及限制非线性项振荡所需的在对流方程中传播正则性的方法。新方法依赖于通过变量加倍直接控制解的正则性，并且需要精确而明确的换位器估计。这种明显截然不同的问题之间的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：具有物理上真实的无界作用力的相互作用粒子系统，或者速度上的随机碰撞，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。研究生被包括在该项目的工作中。

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.