# Conservation of the circulation for the Euler and Euler-Leray equations

## TL;DR

This paper provides a detailed proof of circulation conservation in Euler equations, reformulates it for Euler-Leray equations relevant to self-similar solutions, and discusses implications for their existence.

## Contribution

It offers an explicit proof of circulation conservation and explores its implications for self-similar solutions in Euler-Leray equations.

## Key findings

- Circulation is conserved for Euler solutions.
- Reformulation of circulation conservation for Euler-Leray equations.
- Implications for the existence of self-similar solutions.

## Abstract

It is well-known that the circulation of the velocity field of a fluid along a closed material curve is conserved for any solution of the Euler equation. We offer a slightly more explicit proof of that fact than that generally found in the literature. We then rewrite that property in terms of the rescaled variables and functions leading to the Euler-Leray equations and appropriate for studying self-similar solutions. We finally discuss the implications of the conservation of circulation on the existence of such solutions.