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 \(BC\bigl ([0,T)\,;\,L^{n,\infty }({\mathbb {R}}^n)\bigr )\).

研究了\(L^{n,\infty }({\mathbb {R}}^n)\)中不可压缩Navier-Stokes方程的初值问题。引入 Duhamel 项的真实插值估计，我们在\(BC\bigl ((0,T)\,;\,L^{n,\infty }({ \mathbb {R}}^n)\bigr )\)用于尺度不变类中具有非发散形式的外力。我们观察到温和解变成强解，即它满足\(L^{n,\infty }({\mathbb {R}}^n)\)的临界拓扑中的微分方程，并附加一个条件仅在外力上，即使斯托克斯半群在\(L^{n,\infty }({\mathbb {R}}^n)\)上不是强连续的. 此外，通过局部时间解的存在性和唯一性，我们扩展了解类\(BC\bigl ([0,T)\,;\,L^{n,\infty }({\ mathbb {R}}^n)\bigr )\)。