Well-posedness of the Euler system of gas dynamics
Eduard Feireisl, Maria Lukacova-Medvidova
TL;DR
The paper addresses non-uniqueness in the Euler system for gas dynamics by proposing two principled selection procedures within the framework of dissipative measure-valued solutions. The two-step approach first selects entropy-admissible DMV solutions via maximal entropy production, then isolates a unique turbulent solution by minimizing the mean energy, with the energy defect vanishing at large times; a parallel one-step criterion based on a distance to equilibrium provides an alternative path. The authors establish well-posedness in a measurable, semigroup setting and prove key properties such as DiPerna maximality for Step 1, measurability and convergence results for Step 2, and the asymptotic vanishing of turbulent energy. They also extend the framework to a maximal-dissipativity criterion and discuss numerical implications, edge cases, and the mathematical structure (e.g., Moreau–Yosida regularisation, Mosco convergence). Overall, the work offers a rigorous route to a unique, physically consistent evolution for the Euler system under general initial data.
Abstract
We propose a new two-step selection criterion applicable to the dissipative measure--valued solutions of the Euler system of gas dynamics. The process consists of a successive maximisation of the entropy production rate and the total energy defect, i.e. maximisation of the turbulent energy. If the selected solution is a weak solution of the Euler system, then it is identified in the first step. Solutions selected in the second step are truly measure--valued maximising the energy defect. Accordingly, they are called turbulent solutions. The energy defect of turbulent solutions vanishes with growing time. The selected solutions depend in a Borel--measurable way on the initial data. In particular, they are almost continuously dependent on the initial data.