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 $L^{\mathrm{\infty }}(0,T;L^d({\mathbb{R}}^d))$ when $d\ge 4$, and in $C([0,T];L^d({\mathbb{R}}^d))$ when $d\ge 3$. As for the forced Navier-Stokes equations, when $d\ge 3$ the uniqueness of mild solutions in $C([0,T];L^{d,\mathrm{\infty }}({\mathbb{R}}^d))$ with force $f$ and initial data $u_0$ in some proper Lorentz spaces is known. In this paper we show that for $d\ge 3$, the uniqueness of mild solutions to the forced Navier-Stokes equations in $C((0,T];{\tilde{L}}^{d,\mathrm{\infty }}({\mathbb{R}}^d))\cap L^{\beta }(0,T;{\tilde{L}}^{d,\mathrm{\infty }}({\mathbb{R}}^d))$ for $\beta >2d\mathrm{/}(d-2)$ holds when there is a mild solution in $C([0,T];{\tilde{L}}^{d,\mathrm{\infty }}({\mathbb{R}}^d))$ with the same initial data and force. Here ${\tilde{L}}^{d,\mathrm{\infty }}$ is the closure of $L^{\mathrm{\infty }}\cap L^{d,\mathrm{\infty }}$ with respect to $L^{d,\mathrm{\infty }}$ norm.