Small scale hydrodynamics

The purpose of this work is to offer a brief survey of some of the mathematical methods useful to bridge different levels of description, i.e. from the set-up of classical kinetic theory of gases to hydrodynamics. Most of the standard mathematical techniques, in this field, stem from the seminal Chapman–Enskog expansion, which constitutes an important success in kinetic theory, as it made possible to formally derive the Navier-Stokes hydrodynamics from the Boltzmann equation. Yet, almost a century of effort to extend the hydrodynamic description beyond the Navier-Stokes-Fourier approximation failed even in the case of small deviations around the equilibrium, due to the onset of instabilities which also cause the violation of the H-Theorem. A different route, in kinetic theory, is represented by the recent Invariant Manifold method. The latter technique allows one to restore stability in the hydrodynamic equations, which remain thus applicable even at short length scales, under the hypothesis of validity of the Local Equilibrium condition.