$ L^{p}-L^{q} $ framework of the vorticity criteria for energy conservation of the Euler equations

Qi Zhang; Zhikang Zhang; Xiongbo Zheng; Yongliang Zhou; Qi Zhang; Zhikang Zhang; Xiongbo Zheng; Yongliang Zhou

doi:10.3934/era.2026277

Electronic Research Archive

2026, Volume 34, Issue 9: 6349-6362. doi: 10.3934/era.2026277

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Theory article

$ L^{p}-L^{q} $ framework of the vorticity criteria for energy conservation of the Euler equations

School of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China

Received: 13 March 2026 Revised: 08 June 2026 Accepted: 02 July 2026 Published: 17 July 2026

In this paper, we investigated the energy conservation for the incompressible homogeneous and inhomogeneous Euler equations on a periodic domain of dimensions $n = 2, 3$, enabling for the presence of a vacuum. We established sufficient conditions based on the integrability of the vorticity: If the vorticity belongs to space $L^{p}(0, T; L^{q}(\mathbb{T}^{n}))$ with the exponents $p, q$ satisfying:
$ \quad\quad\quad\quad\quad\left(\frac{1}{2} + \frac{1}{n}\right)\frac{1}{p} + \frac{1}{q} \leq \frac{1}{2} + \frac{1}{n}, \quad p \geq 2, \quad \frac{3n}{n+2}\leq q<n, $
and the density satisfies suitable conditions, then the energy equality holds for almost every $ t\in[0, T] $. Our result extended several existing vorticity-based energy conservation criteria by providing a weaker integrability condition on $\nabla \sqrt{\rho}$.
- Euler equations,
- inhomogeneous Euler equations,
- energy conservation,
- vorticity,
- weak solutions
Citation: Qi Zhang, Zhikang Zhang, Xiongbo Zheng, Yongliang Zhou. $ L^{p}-L^{q} $ framework of the vorticity criteria for energy conservation of the Euler equations[J]. Electronic Research Archive, 2026, 34(9): 6349-6362. doi: 10.3934/era.2026277

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Abstract

In this paper, we investigated the energy conservation for the incompressible homogeneous and inhomogeneous Euler equations on a periodic domain of dimensions $n = 2, 3$, enabling for the presence of a vacuum. We established sufficient conditions based on the integrability of the vorticity: If the vorticity belongs to space $L^{p}(0, T; L^{q}(\mathbb{T}^{n}))$ with the exponents $p, q$ satisfying:

$ \quad\quad\quad\quad\quad\left(\frac{1}{2} + \frac{1}{n}\right)\frac{1}{p} + \frac{1}{q} \leq \frac{1}{2} + \frac{1}{n}, \quad p \geq 2, \quad \frac{3n}{n+2}\leq q<n, $

and the density satisfies suitable conditions, then the energy equality holds for almost every $ t\in[0, T] $. Our result extended several existing vorticity-based energy conservation criteria by providing a weaker integrability condition on $\nabla \sqrt{\rho}$.

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