A note on helicity conservation for compressible Euler equations in a bounded domain with vacuum

TL;DR

This paper investigates conditions under which helicity is conserved for weak solutions of the compressible Euler equations in bounded domains with vacuum, emphasizing boundary regularity and interior conditions.

Contribution

It establishes a sufficient condition for helicity conservation involving interior Besov-VMO regularity and boundary regularity for velocity, vorticity, and density.

Findings

Helicity conservation is guaranteed under specific regularity conditions.

Boundary regularity plays a crucial role in helicity conservation.

The paper extends understanding to compressible flows with vacuum.

Abstract

In this paper, we consider the helicity conservation of weak solutions for the compressible Euler equations in a bounded domain with general pressure law and vacuum. We deduce a sufficient condition for a weak solution conserving the helicity based on the interior Besov-VMO type regularity, the continuous conditions for velocity and vorticity near the boundary, and some regularities for density near vacuum.