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

Yulin Ye

TL;DR

The paper investigates helicity conservation for weak solutions of the compressible Euler equations in a bounded domain with possible vacuum. It derives a sufficient condition for helicity conservation based on interior Besov–VMO regularity for density and velocity, a boundary-continuity flux condition, and regularity of density near vacuum, using a boundary-capturing test function and Constantin–E–Titi type commutator estimates. The authors extend prior results from periodic domains to bounded domains with vacuum by first obtaining a local helicity balance for mollified solutions and then upgrading to a global conservation via a boundary cutoff, proving that $\int_{\Omega} \omega\cdot v\,dx$ remains constant in time. This work contributes a rigorous criterion for helicity preservation in compressible flows with vacuum, with implications for the study of topological aspects and turbulence in bounded domains.

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.

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

TL;DR

The paper investigates helicity conservation for weak solutions of the compressible Euler equations in a bounded domain with possible vacuum. It derives a sufficient condition for helicity conservation based on interior Besov–VMO regularity for density and velocity, a boundary-continuity flux condition, and regularity of density near vacuum, using a boundary-capturing test function and Constantin–E–Titi type commutator estimates. The authors extend prior results from periodic domains to bounded domains with vacuum by first obtaining a local helicity balance for mollified solutions and then upgrading to a global conservation via a boundary cutoff, proving that remains constant in time. This work contributes a rigorous criterion for helicity preservation in compressible flows with vacuum, with implications for the study of topological aspects and turbulence in bounded domains.

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.

Paper Structure

This paper contains 3 sections, 8 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Notations and some auxiliary lemmas
  3. Helicity conservation for the compressible Euler equations

Key Result

Theorem 1.1

Let the pair $(\rho, v)$ be a weak solution of the compressible Euler equations CEuler in the Definition of ceulerdefi and $\omega\in C([0,T];L^{\frac{3}{2}}(\Omega))$, $v\in C([0,T];L^{3}(\Omega))$. Assume that for some positive constant $c,r_1,r_2,r_3$ and some sufficiently small constant $\delta>0$ such that $\frac{1}{r_1}+\frac{1}{r_2}=\frac{2}{3}$ and $\frac{1}{r_1}+\frac{1}{r_3}=\frac{1}{3}

Theorems & Definitions (18)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • proof
  • ...and 8 more