The limit of the Boltzmann equation to the Euler equations for Riemann problems (Q2843468)

It has been believed since the second half of the 19th century that the Boltzmann kinetic equation serves as a good starting point to deduce the equations of motion of Euler or even the Navier-Stokes equations. The open questions were at that time connected with the particles of gases, since only the most elementary views on the properties of these particles were supposed. This is symbolized by the introduction of a physical parameter called the Knudsen number, which had to balance the various terms of the kinetic equation. This was evident, since at that time, from the contributions of D. Hilbert and later other researchers it has been stated that we do not know so much on the nature of interaction of the atoms.NEWLINENEWLINE This report, after summarizing shortly the historic background, starts with the analysis of the idea of Riemann on the one-dimensional isentropic flow of gas dynamics. Following his idea, the authors re-introduce the three characteristic wave types: the compressive shock, the expanding rarefaction shock and the contact discontinuity, this last one having some diffusive structure, while the others represent some mass and other properties. The two ``waves'' with their hyperbolic approximative structure help the authors to prove rigorously that there exists a family of solutions to the Boltzmann equation that converges to the Maxwellian distribution as determined by Riemann's solution, consisting of the above three parts, when the Knudsen number tends to zero.NEWLINENEWLINE The long paper has the following structure. After the introductory re-capitulating chapter, the approximate wave patterns are discussed (rarefaction wave, hyperbolic wave No. I and No. II, and the superposition of waves).NEWLINENEWLINE Then comes the proof of the main results, after the reformulation of the starting points. Energy considerations help to trace the proofs and the conclusion of the paper.