This thesis is devoted to the mathematical study of the Boltzmann equation, a fundamental model of collisional kinetic theory, and contains two distinct contributions to the subject.

Firstly, we establish a rigorous demonstration of the hydrodynamic convergence of the Boltzmann equation towards a Navier-Stokes-Fourier system under the presence of long-range interactions. This convergence is obtained by letting the Knudsen number tend to zero and has been known to hold at least formally for over a decade. It is only recently that a fully rigorous mathematical derivation of this hydrodynamic limit was discovered. However, these results failed to encompass almost all physically relevant collision kernels due to the cutoff assumption, which requires that the cross sections be integrable. Indeed, as soon as long-range intermolecular forces are present, non-integrable collision kernels have to be considered because of the enormous amount of grazing collisions in the gas. In this long-range setting, the Boltzmann operator becomes a singular integral operator and the known rigorous proofs of hydrodynamic convergence just don't carry over to that case. In fact, the DiPerna-Lions solutions don't even make sense in this situation and the relevant global solutions to the Boltzmann equation are the so-called renormalized solutions with a defect measure developed by Alexandre and Villani. Our work overcomes the new mathematical difficulties coming from long-range interactions by proving the hydrodynamic convergence of the Alexandre-Villani solutions towards the Leray solutions.

Secondly, we develop a new theory of existence of global solutions to the Boltzmann equation for small initial data, which we name mild solutions in analogy with the mild solutions for the Navier-Stokes equations. The existence comes as a result of the study of the competing phenomena of dispersion, due to the transport operator, and of singularity formation, due to the nonlinear Boltzmann collision operator. It is the joint use of the so-called dispersive estimates with new convolution inequalities on the gain term of the collision operator that allows to obtain uniform bounds on the solutions and thus demonstrate the existence of solutions.