From Boltzmann’s Equation to the Incompressible Navier–Stokes–Fourier System with Long-Range Interactions

Abstract

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 decades. It is only more 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 a 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 number 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 simply do not carry over to that case. In fact, the DiPerna–Lions renormalized solutions do not 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 the consideration of long-range interactions by proving the hydrodynamic convergence of the Alexandre–Villani solutions towards the Leray solutions.