The Principal Investigator (PI) proposes to continue working on some asymptotic problems coming from Fluid Mechanics, Gas Dynamics and Quantum Mechanics. These asymptotic problems arise when a dimensionless parameter (such as the Reynolds number, the Mach number or a time scaling) goes to zero or to infinity. In studying these problems, many mathematical difficulties arise. These difficulties are mainly due to the change of the type of the equations, the presence of many temporal and spatial scales, the presence of resonances, the presence of boundary layers ... Many tools have been developed to circumvent these difficulties such as the introduction of different types of measures to describe the defect of strong convergence, the use of compensated compactness type arguments, the use of averaging lemma, the use of energy methods and relative entropy methods ... Of course, many other tools should be developed in the future. More specifically the PI plans to continue working on the hydrodynamic limits of the Boltzmann equation when the Knudsen number goes to 0 especially in bounded domains and relate this to some of the known results in the compressible-incompressible limit. He also plans to look at the limit when the Weber number goes to infinity in the water wave problem.
The PI proposes to study some asymptotic problems coming from Fluid Mechanics, Gas Dynamics and Quantum Mechanics. The study of these asymptotic problems is very important to get a better understanding about the behavior of complicated systems in different limiting cases. This also allows to have a better understanding of the real physical phenomenon taking place. Moreover, it provides a way of giving rigorous derivations of different physical models. It also gives a better knowledge about the domain of validity of each simplifying model. This is very important for engineers and physicists who are looking for the simplest model that captures the phenomenon to implement numerically or to apply in real life.
