A remark on the uniqueness of Kozono–Nakao’s mild $$L^3$$ L 3 -solutions on the whole time axis to the Navier–Stokes equations in unbounded domains

This paper is concerned with the uniqueness of Kozono–Nakao’s bounded continuous $$L^{3}$$ L 3 -solutions on the whole time axis to the Navier–Stokes equations in 3-dimensional unbounded domains. When $$\Omega $$ Ω is an unbounded domain, it is known that a small solution in $$BC({\mathbb {R}};L^{3,\infty })$$ B C ( R ; L 3 , ∞ ) is unique within the class of solutions which have sufficiently small $$L^{\infty }({\mathbb {R}}; L^{3,\infty })$$ L ∞ ( R ; L 3 , ∞ ) -norm. There is another type of uniqueness theorem. Farwig, Nakatsuka and the author (2015) showed that if two solutions exist for the same force f, one is small and if other one satisfies the precompact range condition (PRC), then the two solutions coincide. Since time-periodic solutions satisfy (PRC), this uniqueness theorem is applicable to time-periodic solutions. On the other hand, there exist many solutions which do not satisfy (PRC). In this paper, by assuming the boundedness of the $$L^r$$ L r -norm for some $$1