Flows, Waves, and their Asymptotic Stability

Introduction

Equations of waves and flows are used extensively in physics and biology to describe phenomena ranging from the flow past an airfoil to the collective motion of cells and the motion of water surfaces. A major issue is to explain how the propagation through space and the concentration to various scales can emerge from these mathematical models.

Background

Fundamental progress has been made since the beginning of the millennium around the role played by specific solutions that either propagate or shrink while keeping the same shape, such as solitary waves, for example. These specific solutions are the key to understanding the global dynamics.

Project Goals

The goal of this project is to push forward the current knowledge on:

1. Their stability
2. Their emergence over time
3. The dynamics they are responsible for in several equations

The FloWAS project will study seemingly unrelated models, whose solutions in fact display remarkably close behaviors.

Research Areas

Fluid Mechanics

First, we aim at describing how a thin layer of fluid can detach off a boundary and be ejected away in a stream. This is a key phenomenon to understand the drag exerted on moving objects. For this, we will study singular solutions of the unsteady Prandtl system of fluid mechanics.

Bacterial Movement

Second, we will study concentration phenomena arising in the movement of bacteria. For that, we will consider nonlinear structures appearing in the Keller-Segel system: how they can collapse and how they can interact.

Wave Dynamics

Third, we will consider how, from initially disordered wave packets, order appears over time and traveling waves emerge. This study will be made on the critical wave equation. Applications to weak wave turbulence will be pursued.

Conclusion

Describing all these phenomena lies at the frontier of current research, and we expect applications to a wide range of models.