We revisit the well-known work of K. Masuda in 1984 on the weak–strong uniqueness of $L^{\infty }{L}^3$ Leray–Hopf weak solutions of Navier–Stokes equation. We modify the argument, and extend the uniqueness result to the scaling critical anisotropic Lebesgue space with mixed-norms. As a consequence, our results cover the class of initial data and solutions which may be singular or decay with different rates along different spatial variables. The result relies on the establishment of several refined properties of solutions of the Stokes and Navier–Stokes equations in mixed-norm Lebesgue spaces which seem to be of independent interest.

我们重温了 K. Masuda 在 1984 年关于弱-强唯一性的著名著作 ${升}^{\infty }{升}^3$Navier-Stokes 方程的 Leray-Hopf 弱解。我们修改参数，并将唯一性结果扩展到具有混合范数的缩放临界各向异性 Lebesgue 空间。因此，我们的结果涵盖了初始数据和解决方案的类别，这些数据和解决方案可能是奇异的，也可能是沿不同空间变量以不同速率衰减的。结果依赖于在混合范数 Lebesgue 空间中建立 Stokes 和 Navier-Stokes 方程解的几个精细性质，这似乎是独立的兴趣。