Well-posedness of the Euler system of gas dynamics

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.

💡 Research Summary

The paper addresses the longstanding ill‑posedness of the compressible Euler system for a gas, where classical solutions develop shocks and infinitely many weak solutions satisfy the entropy inequality. To overcome this, the authors work within the framework of dissipative measure‑valued (DMV) solutions, which are parametrised probability measures together with a concentration defect measure. DMV solutions arise naturally as limits of vanishing‑viscosity approximations or consistent numerical schemes and satisfy the continuity, momentum, entropy, and energy compatibility relations in a weak, measure‑theoretic sense.

The central contribution is a two‑step selection principle that singles out a unique, physically admissible solution for any given initial data (ρ₀, m₀, S₀) and total energy E₀. In the first step the functional

 F_S