This paper is concerned with the problem of energy conservation for the two dimensional inhomogeneous Euler equations, with an emphasis on the vorticity \(\omega =\mathrm{curl}u\) of the flow. In particular, two types of sufficient conditions are obtained. The first one assumes \(L^p\)-regularity on the spatial gradient of the density \(\nabla \rho \) and the vorticity \(\omega \). The second one removes the regularity condition on \(\nabla \rho \) while requires certain time regularity of \(\omega \). Furthermore, we phrase the energy spectrum in terms of the Littlewood–Paley decomposition and show that the energy flux \(\Pi _q\) vanishes as the dyadic exponent \(q\rightarrow \infty \).

中文翻译：

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涡度具有正则性的二维密度相关Euler方程的能量守恒

本文关注二维非齐次Euler方程的能量守恒问题，重点是流动的涡度\（\ omega = \ mathrm {curl} u \）。特别地，获得两种类型的充分条件。第一个假设在密度\（\ nabla \ rho \）和涡度\（\ omega \）的空间梯度上具有\（L ^ p \） -正则性。第二个规则删除\（\ nabla \ rho \）上的正则条件，而需要一定时间的\（\ omega \）。此外，我们用Littlewood–Paley分解的形式来表达能量谱，并表明能量通量\（\ Pi _q \）随着二进指数\（q \ rightarrow \ infty \）消失。