This paper is concerned with the energy conservation for the weak solutions to the 3D incompressible Navier–Stokes–Cahn–Hilliard system. It is shown that just $v\in L^p(0,T;L^q\left({\mathbb{T}}^3\right))$ with $\frac{2}{p}+\frac{2}{q}=1,(q\ge 4)$ or with $\frac{1}{p}+\frac{3}{q}=1,(3<q<4)$ implies the energy equality of weak solutions of these equations, which is a parallel of classical Shinbrot’s condition for energy equality in the 3D Navier–Stokes equations. Moreover, we also show that if $\nabla v\in L^p(0,T;L^s\left({\mathbb{T}}^3\right))$ with $\frac{1}{p}+\frac{6}{5s}=1,(s\ge \frac{9}{5})$, then the energy equality also holds, which improves the recent results in Liang et al. (2020).

中文翻译：

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3D Navier-Stokes-Cahn-Hilliard 系统弱解的能量守恒

本文关注的是 3D 不可压缩 Navier-Stokes-Cahn-Hilliard 系统弱解的能量守恒。表明只是$v\in {升}^{磷}(0,吨;{升}^q\left({\mathbb{吨}}^3\right))$ 和 $\frac{2}{磷}+\frac{2}{q}=1,(q\ge 4)$ 或与 $\frac{1}{磷}+\frac{3}{q}=1,(3<q<4)$暗示这些方程弱解的能量相等，这与 3D Navier-Stokes 方程中能量相等的经典 Shinbrot 条件平行。此外，我们还表明，如果$\nabla v\in {升}^{磷}(0,吨;{升}^{秒}\left({\mathbb{吨}}^3\right))$ 和 $\frac{1}{磷}+\frac{6}{5秒}=1,(秒\ge \frac{9}{5})$，那么能量等式也成立，这改善了梁等人最近的结果。(2020)。