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

Regularity of renormalized solutions in the Boltzmann equation with long‐range interactions

“…We therefore take special care in detailing our hypotheses on the collision kernel. We emphasize that our assumptions encompass the case of inverse power kernels (1.5)-(1.6) and are compatible with the natural assumptions made in the noncutoff theories from [2,3,4].…”

“…We wish now to explain how our main theorem, Theorem 2.1, provides a crucial estimate that is used in [3,4] to justify the rigorous hydrodynamic limit. For the sake of simplicity, we consider here that the spatial domain is merely the Ddimensional torus T D .…”

“…We detail now the alternative strategy used to prove the nonlinear compactness estimate (4.14) in the noncutoff setting (we refer the reader to [3,4] for all the details on that matter) built on the three following properties, which will, respectively, replace items (2), (3), and (4) from the above cutoff strategy: …”

“…However, these results still failed to extend to the noncutoff kernels, as the known rigorous proofs of convergence just don't carry over to that case. It is in [3,4] that the first author, building upon the aforementioned works on the cutoff setting, finally established the incompressible Navier-Stokes-Fourier hydrodynamic limit of the Boltzmann equation in the presence of long-range interactions, thus treating all physically relevant kernels.…”

“…For simplicity, we only consider the case of the inverse power potential kernels (1.5)-(1.6), and refer to [3,4] for broader assumptions on general nonintegrable cross sections. In any case, the collision kernel should always at least satisfy assumptions (2.1), (2.3), and (2.4) in order to validly apply Theorem 2.1, which holds for the inverse power kernels (1.5)-(1.6).…”