L_p − L_q framework of the vorticity criteria for energy conservation of the Euler equations

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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 $L^{p}\left(0, T ; L^{q}\left(\mathbb{T}^{n}\right)\right)$ with the exponents p,q satisfying:

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

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

Keywords

Euler equations / inhomogeneous Euler equations / energy conservation / vorticity / weak solutions

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