# Dissipative solutions and semiflow selection for the complete Euler system

## TL;DR

This paper introduces a dynamical systems approach to select physically relevant solutions for the complete Euler system, ensuring consistency with admissibility criteria and stability of stationary states.

## Contribution

It proposes a novel selection method based on dissipative solutions that satisfies the semiflow property and respects entropy maximization and weak-strong uniqueness.

## Key findings

- Selection of a unique solution system for the Euler equations.
- Ensures strong solutions are always selected when they exist.
- Includes stationary states in the solution set, maintaining stability.

## Abstract

To circumvent the ill-posedness issues present in various models of continuum fluid mechanics, we present a dynamical systems approach aiming at selection of physically relevant solutions. Even under the presence of infinitely many solutions to the full Euler system describing the motion of a compressible inviscid fluid, our approach permits to select a system of solutions (one trajectory for every initial condition) satisfying the classical semiflow property. Moreover, the selection respects the well accepted admissibility criteria for physical solutions, namely, maximization of the entropy production rate and the weak--strong uniqueness principle. Consequently, strong solutions are always selected whenever they exist and stationary states are stable and included in the selection as well. To this end, we introduce a notion of dissipative solution, which is given by a triple of density, momentum and total entropy defined as expectations of a suitable measure--valued solution.