The incompressible Euler limit

The aim of this chapter is to describe the state of the art about the incompressible Euler limit of the Boltzmann equation, which is not so complete as the incompresible Navier-Stokes limit presented in the previous chapter.

Due to the lack of regularity estimates for inviscid incompressible models, the convergence results describing the incompressible Euler asymptotics of the Boltzmann equation require additional regularity assumptions on the solution to the target equations.

Furthermore, the relative entropy method leading to these stability results controls the convergence in a very strong sense, which imposes additional conditions either on the solution to the asymptotic equations (“well-prepared initial data“), or on the solutions to the scaled Boltzmann equation (namely some additional non uniform a priori estimates giving in particular the local conservation of momentum and energy).