# On convergence of approximate solutions to the compressible Euler system

## TL;DR

This paper investigates the convergence behavior of approximate solutions to the compressible Euler system, showing they either converge strongly or do not form valid solutions, contrasting with the incompressible case.

## Contribution

It establishes a dichotomy for approximate solutions to the compressible Euler system, linking convergence to the triviality of turbulent defect measures.

## Key findings

- Approximate solutions either converge strongly or are not solutions.
- The approach uses differential equations for turbulent defect measures.
- Contrasts with the incompressible Euler case where weak convergence is possible.

## Abstract

We consider a sequence of approximate solutions to the compressible Euler system admitting uniform energy bounds and/or satisfying the relevant field equations modulo an error vanishing in the asymptotic limit. We show that such a sequence either (i) converges strongly in the energy norm, or (ii) the limit is not a weak solution of the associated Euler system. This is in sharp contrast to the incompressible case, where (oscillatory) approximate solutions may converge weakly to solutions of the Euler system. Our approach leans on identifying a system of differential equations satisfied by the associated turbulent defect measure and showing that it only has a trivial solution.