# Solution semiflow to the isentropic Euler system

## TL;DR

This paper introduces a new framework for the isentropic Euler system using Markov semigroups, establishing a measurable solution semiflow that ensures well-posedness despite the system's inherent ill-posedness.

## Contribution

It proposes a novel approach based on dissipative solutions and Markov semigroups to address well-posedness issues in the multidimensional isentropic Euler system.

## Key findings

- Existence of a Borel measurable solution semiflow.
- Solutions coincide with strong solutions when they exist.
- Solutions minimize energy among dissipative solutions.

## Abstract

It is nowadays well understood that the multidimensional isentropic Euler system is desperately ill--posed. Even certain smooth initial data give rise to infinitely many solutions and all available selection criteria fail to ensure both global existence and uniqueness. We propose a different approach to well--posedness of this system based on ideas from the theory of Markov semigroups: we show the existence of a Borel measurable solution semiflow. To this end, we introduce a notion of dissipative solution which is understood as time dependent trajectories of the basic state variables - the mass density, the linear momentum, and the energy - in a suitable phase space. The underlying system of PDEs is satisfied in a generalized sense. The solution semiflow enjoys the standard semigroup property and the solutions coincide with the strong solutions as long as the latter exist. Moreover, they minimize the energy (maximize the energy dissipation) among all dissipative solutions.