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

Abstract

The initial value problem of the incompressible Navier–Stokes equations in \(L^{n,\infty }({\mathbb {R}}^n)\) is investigated. Introducing the real interpolation estimates for the Duhamel terms, we construct global and local in time mild (and strong) solutions in \(BC\bigl ((0,T)\,;\,L^{n,\infty }({\mathbb {R}}^n)\bigr )\) for external forces with non-divergence form in the scale invariant class. We observe that a mild solution becomes the strong solution, i.e., it satisfies the differential equation in the critical topology of \(L^{n,\infty }({\mathbb {R}}^n)\) with an additional condition only on the external force, even though the Stokes semigroup in not strongly continuous on \(L^{n,\infty }({\mathbb {R}}^n)\). Furthermore, via the existence and the uniqueness of local in time solutions, we extend the uniqueness theorem within the solution class \(BC\bigl ([0,T)\,;\,L^{n,\infty }({\mathbb {R}}^n)\bigr )\).