We discuss recent developments around Hilbert’s Sixth Problem about the passage from kinetic models to macroscopic fluid equations. We employ the technique of slow spectral closure to rigorously establish the existence of hydrodynamic manifolds in the linear regime and derive new non-local fluid equations for rarefied flows independent of Knudsen number. We show the divergence of the Chapman–Enskog series for an explicit example and apply machine learning to learn the optimal hydrodynamic closure from DSMC data. The new dynamically optimal constitutive laws are applied to a rarefied flow problem and we discuss the classical problem of the number of macroscopic rarefied fluid fields from a data-driven point of view.

This talk focuses on a kinetic model for polyatomic gases, where the microscopic internal energy and kinetic energy are independently conserved during collision processes, known as resonant collisions. Such behavior is observed, for example, in some selectively excited polyatomic molecules. The peculiar structure of resonant collision rules enables the tensorization of the problem, allowing for the separate treatment of internal energy contributions. We describe the main properties of the model and study some questions on the macroscopic dynamics of the system.