Analytical validation of the helicity conservation for the compressible Euler equations

TL;DR

This paper rigorously proves that weak solutions of the compressible Euler equations in certain function spaces conserve helicity, extending the understanding of helicity preservation in fluid dynamics.

Contribution

It establishes helicity conservation for weak solutions of the compressible Euler equations in Onsager-type spaces, generalizing previous smooth solution results.

Findings

Helicity is conserved for weak solutions in specific Besov spaces.

Results extend to incompressible Euler and surface quasi-geostrophic equations.

Provides a mathematical foundation for helicity conservation in weak solutions.

Abstract

In [25], Moffatt introduced the concept of helicity in an inviscid fluid and examined the helicity preservation of smooth solution to barotropic compressible flow. In this paper, it is shown that the weak solutions of the above system in Onsager type spaces $\dot{B}^{1/3}_{p,c(\mathbb{N})}$ guarantee the conservation of the helicity. The parallel results of homogeneous incompressible Euler equations and the surface quasi-geostrophic equation are also obtained.