Statistical solutions to the Euler system of gas dynamics
Eduard Feireisl
TL;DR
The paper addresses the ill-posedness of the Euler system for a compressible perfect fluid by formulating a statistical framework built on dissipative (measure-valued) solutions. It introduces a single-step selection criterion based on minimizing the Bregman distance to the maximal entropy equilibrium, ensuring a unique admissible dissipative solution whose energy defect vanishes as time grows. From this, it constructs a Markov semigroup on probability measures—i.e., a statistical solution—via pushforward along the solution semigroup, and proves its key regularity and convergence properties. The results connect energetic and entropic considerations to long-time dynamics and turbulence modeling, suggesting that the selected solution asymptotically approaches a weak Euler solution and providing a rigorous basis for evolution of distributions in turbulent flows. Open questions remain about whether the selected solution is always a weak solution in the classical sense and how the framework extends to alternative selection functionals.
Abstract
We consider the (complete) Euler system describing the motion of a compressible perfect fluid. We propose a platform suitable for constructing the statistical solutions. The main ingredients of our approach include: 1. The concept of dissipative (measure{valued) solution to the Euler system. 2. A single step selection procedure based on minimizing the Bregman divergence of a given solution to the maximal entropy equilibrium. 3. A construction of a Markov semigroup via push forward measures.