# On regularity of the Euler equations in fluid dynamics

## TL;DR

This paper demonstrates that solutions to the inviscid vorticity equation remain regular and unique for smooth initial data, while primitive Euler equations are ill-posed, implying no anomalous energy dissipation in inviscid flows.

## Contribution

It clarifies the regularity of vorticity-based solutions and highlights issues with primitive formulations, providing new insights into energy dissipation and flow behavior.

## Key findings

- Solutions to the vorticity equation are regular and unique for smooth initial data.
- Primitive Euler equations are ill-posed due to passive pressure.
- Anomalous energy dissipation cannot occur in inviscid flows.

## Abstract

We assert that the solutions to the Cauchy problem of the inviscid vorticity equation remain regular and unique for any smooth initial data of finite energy. However, the primitive formulation of the Euler equations is not well-posed, due to the passive pressure. One of the implications is that the anomalous energy dissipation, anticipated by Onsager (1949), cannot occur in inviscid flows. In the complete absence of viscous effects, the ultimate accumulation of enstrophy in sealed domains is bound to become arbitrarily excessive, if there is a sustained supply of shears and strains.