This project is concerned with the study of problems coming from fluid mechanics, gas dynamics, and plasma physics. The principal investigator with his collaborators P. Germain and J. Shatah develops the method of space-time resonances, which combines the standard notion of resonances with the study of the spatial localization of solutions. One application of the method is the global existence and scattering for water waves and for capillary waves. He studies asymptotic problems in fluid mechanics, especially those giving rise to boundary layers: zero viscosity limit and compressible-incompressible limit in the presence of a free boundary, and the hydrodynamic limit of the Boltzmann equation in a bounded domain. Understanding the properties of the boundary layer is important in many applications. He investigates the homogenization of elliptic operators in the presence of oscillating boundary data. He also studies global existence of weak solutions for some non-Newtonian fluids (viscoelastic) and especially polymeric liquids. These systems require a coupling between a fluid equation and the Fokker-Planck equation for the polymers.
The investigator studies the behavior of complicated systems with different boundary conditions and in different limiting cases. These studies provide an improved understanding of the real physical phenomena taking place and of the domain of validity of each simplifying model. This is important for engineers and physicists who look for the simplest model that captures the phenomena, to implement numerically or to apply in real life. For instance, water waves problems are important to understand tsunamis and rogue waves. Moreover, the study of viscoelastic fluids and especially polymeric liquids (egg white, blood, or dough for example) is important in many industrial applications such as food processing and is of great interest in many branches of applied physics, chemistry, and biology. This project includes the training of graduate students in mathematics and physics.