Uniqueness of mild solutions to the Navier-Stokes equations in weak-type $L^d$ space

This paper deals with the uniqueness of mild solutions to the forced or unforced Navier-Stokes equations in the whole space. It is known that the uniqueness of mild solutions to the unforced Navier-Stokes equations holds in $L^{\infty}(0,T;L^d(\mathbb{R}^d))$ when $d\geq 4$, and in $C([0,T];L^d(\mathbb{R}^d))$ when $d\geq3$. As for the forced Navier-Stokes equations, when $d\geq3$ the uniqueness of mild solutions in $C([0,T];L^{d,\infty}(\mathbb{R}^d))$ with force $f$ and initial data $u_{0}$ in some proper Lorentz spaces is known. In this paper we show that for $d\geq3$, the uniqueness of mild solutions to the forced Navier-Stokes equations in $ C((0,T];\widetilde{L}^{d,\infty}(\mathbb{R}^d))\cap L^\beta(0,T;\widetilde{L}^{d,\infty}(\mathbb{R}^d))$ for $\beta>2d/(d-2)$ holds when there is a mild solution in $C([0,T];\widetilde{L}^{d,\infty}(\mathbb{R}^d))$ with the same initial data and force. Here $\widetilde{L}^{d,\infty}$ is the closure of ${L^{\infty}\cap L^{d,\infty}}$ with respect to $L^{d,\infty}$ norm.