# On solvability and ill-posedness of the compressible Euler system subject to stochastic forces

## TL;DR

This paper demonstrates that the stochastic compressible Euler system is ill-posed in the weak solution class, with infinitely many solutions emerging from the same initial data, using convex integration methods.

## Contribution

It adapts convex integration to show ill-posedness of the stochastic Euler system with multiple solutions from identical initial conditions.

## Key findings

- Existence of infinitely many solutions for the same initial data.
- Ill-posedness in the class of weak solutions.
- Solutions are strong in the probabilistic sense.

## Abstract

We consider the (barotropic) Euler system describing the motion of a compressible inviscid fluid driven by a stochastic forcing. Adapting the method of convex integration we show that the initial value problem is ill-posed in the class of weak (distributional) solutions. Specifically, we find a sequence $\tau_M \to \infty$ of positive stopping times for which the Euler system admits infinitely many solutions originating from the same initial data. The solutions are weak in the PDE sense but strong in the probabilistic sense, meaning, they are defined on an {\it a priori} given stochastic basis and adapted to the driving stochastic process.