On Masuda uniqueness theorem for Leray–Hopf weak solutions in mixed-norm spaces

Abstract

We revisit the well-known work of K. Masuda in 1984 on the weak–strong uniqueness of L∞L3 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.