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

Jabin

DMS-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.