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The initial value problem of the incompressible Navier-Stokes equations withnon-zero forces in $L^{n,\infty}(\mathbb{R}^n)$ is investigated. Even thoughthe Stokes semigroup is not strongly continuous on$L^{n,\infty}(\mathbb{R}^n)$, with the qualitative condition for the externalforces, it is clarified that the mild solution of the Naiver-Stokes equationssatisfies the differential equations in the topology of$L^{n,\infty}(\mathbb{R}^n)$. Inspired by the conditions for the forces, wecharacterize the maximal complete subspace in $L^{n,\infty}(\mathbb{R}^n)$where the Stokes semigroup is strongly continuous at $t=0$. By virtue of thissubspace, we also show local well-posedness of the strong solvability of theCauchy problem without any smallness condition on the initial data in thesubspace. Finally, we discuss the uniqueness criterion for the mild solutionsin weak Lebesgue spaces by the argument by Brezis.

Remark on the strong solvability of the Navier-Stokes equations in the weak $L^n$ space