Incompressible hydrodynamics as a limit of the boltzmann equation

This paper is devoted to proofs of convergence for the incompressible hydrodynamic limits of the Boltzmann equation. It focuses on methods such as entropy control, dissipation control and velocity averaging that allow to treat the case of global weak solutions of the Boltzmann equation (due to DiPema-Lions) converging to global weak solutions of the Navier-Stokes equation (due to Leray). In particular, it shows that the DiPerna-Lions version of the H theorem yields the Leray energy inequality for the Navier-Stokes equation in the limit.