# What Can Mathematical Analysis Reveal About The Behavior Of A Fluid? The study of flows or fluid mechanics began since antiquity for everyday applications such as irrigation in agriculture. Several innovative and complex irrigation systems have been discovered in ancient cities. The inclusion of mathematics has revolutionized the fluid mechanics owing to the works of Daniel Bernoulli, Jean Rond d’Alembert, and Leonhard Euler. The introduction by Henry Navier in 1820 of the concept of viscosity, allowed George Gabriel Stokes to introduce the Navier-Stokes equations which will mark the whole history of fluid mechanics.

The Navier-Stokes equations are nonlinear partial differential equations that describe the motion of gases and most liquids. Joseph Boussinesq proposes an alternative model, which also takes into account the temperature variations. Indeed, a fluid particle is driven by the flow, but also exchanges heat with the surrounding environment and undergoes thermal expansion which has an influence on its motion. For instance, a fluid particle heated at the base expands and rises under the effect of buoyancy. Arriving at the top and in contact with a cold source, the fluid particle cools, contracts and creates a return heat transfer. These phenomena create currents in the fluid called convection movements.

The Boussinesq system is a system of partial differential equations that describes the coupling between fluid flow and temperature variations. For a given initial configuration of speed and temperature, the resolution of the system provides information on the flow, and also on the evolution of temperature at any point of the fluid.

Our work provides a mathematical analysis of this system of equations and in particular the behavior of its solutions in the long run. More precisely, we try to answer the following questions: Can we construct solutions to this system that are defined for all times? How does the flow behave when time goes to infinity? What are the conserved quantities? Is there any quantity that could become indefinitely large over time?

In our article, we expose several interesting conclusions. We were able to build suitable solutions for small initial data. But we also discovered a very interesting property of the Boussinesq system. In particular, we have shown that in an unbounded domain, if the initial temperature has non zero mean, the energy of the system can become indefinitely large (blow up) over time. In addition, we provide a quantified measure of the speed at which this phenomenon occurs. This counter-intuitive result raises many questions about the Boussinesq model and its limits and may help to understand it more deeply.

These findings are described in the article entitled A Short Proof of the Large Time Energy Growth for the Boussinesq System, published in the Journal of Nonlinear Science. This work was led by Charafeddine Mouzouni & Lorenzo Brandolese from Institut Camille Jordan.  