# An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations

## TL;DR

This paper establishes a threshold of Besov regularity for compressible Euler solutions, below which entropy and energy dissipation occur, linking turbulence phenomena to mathematical regularity conditions.

## Contribution

It proves a singularity theorem for compressible Euler equations, connecting solution regularity with entropy conservation and energy cascade, and relates Navier-Stokes limits to Euler solutions.

## Key findings

- Entropy is conserved above the Besov regularity threshold.
- Energy cascade vanishes for solutions with sufficient regularity.
- Stationary shocks satisfy the theorem's conditions, demonstrating sharpness.

## Abstract

We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier-Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy dissipation, with both vanishing when solutions are above the critical degree of Besov regularity. Stationary, planar shocks in Euclidean space with an ideal-gas equation of state provide simple examples that satisfy the conditions of our theorems and which demonstrate sharpness of our $L^3$-based conditions. These conditions involve space-time Besov regularity, but we show that they are satisfied by Euler solutions that possess similar space regularity uniformly in time.