On the Strong Solutions to the Navier–Stokes System with Excessively Singular External Forces

The initial value problem for the incompressible Navier–Stokes system on the whole space Rⁿ is considered, where n ≥ 2. By assuming that the initial data a ∈ (Ḃ^_ (Rⁿ) ) ⁿ and the external forces f ∈ L^ ( (0, ∞) ; (Ḣ^ (Rⁿ) ) ⁿ) with n _ (0, T; ( ( (Ḣ^ + Ḣ^) (Rⁿ) ) ⁿ). Moreover, if 2 - 2n/r ≤ s) (Rⁿ) ) ⁿ). The existence of global strong solutions u under the smallness conditions of a and f is also shown. Our results may be regarded as an improved version compared with those of Kozono and Shimizu Math. Nachr. , 291 (2018), pp. 1781-1800; J. Funct. Anal. , 276 (2019), pp. 896-931 in the sense that the lower bound of the index s is extended. In particular, global strong solutions u may be constructed even if the external forces f are given as a form of the double layer potential.