This proposal is about the study of some asymptotic problems in Fluid Mechanics and Gas Dynamics. Asymptotic problems arise when a dimensionless parameter epsilon goes to zero in an equation describing the motion of some type of fluid (or any other physical system). Many mathematical problems are encountered when we try to justify the passage to the limit, which are mainly due to the change of the type of the equations, the presence of many spatial and temporal scales, the presence of boundary layers (we can no longer impose the same boundary conditions for the initial system and the limit one), the presence of oscillations in time at high frequency .... In this Proposal, the PI intends to study (among other problems) the hydrodynamic limit of the Boltzmann equation, the compressible-incompressible limit,
These asymptotic problems allow us to get simpler models at the limit, due to the fact that we usually have fewer variables or (and) fewer unknowns. This simplifies the numerical simulations and improves our understanding of the prevailing phenomenon when the parameter is small, in fact, instead of solving the initial system, we can solve the limit system and then add a corrector.
