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This paper develops L p - L q decay estimates of the gradient of the Stokes semigroup (T (t)) t ≥ 0 generated by the negative of the Stokes operator in exterior Lipschitz domains Ω ⊂ R n , n ≥ 3. More precisely, the L p - L q estimates of ∇ T (t) with optimal rates are proved if p and q satisfy | 1 / p − 1 / 2 | < 1 / (2 n) + ε , | 1 / q − 1 / 2 | < 1 / (2 n) + ε , and p ≤ q ≤ n with some ε > 0 , which should be useful for the study of stability of the several physically relevant flows. As an application, we obtain the existence theorem of global-in-time strong solutions to the three-dimensional Navier–Stokes equations in the critical space L ∞ (0 , ∞ ; L σ 3 (Ω)) provided that the initial velocity is small in the L 3 -norm. [ABSTRACT FROM AUTHOR]

[Display omitted] • We discovered a ricefish species from a river in central Sulawesi. • This river shares estuarine waters with another river where its sister species occurs. • These two species are genetically isolated despite freshwater drainage connectivity. • Coalescent-based demographic inference demonstrated that they are of lake-origin. • They are probably relic species left in these rivers due to lost dispersal ability. Sulawesi is a biodiversity hotspot for ricefishes (Adrianichthyidae), with many species endemic to the central part of this island in single ancient lakes or lake systems. Frequent vicariance by lake fragmentation since the Pliocene may be largely responsible for diversification in this family. In this study, we demonstrate that not only lacustrine species but also riverine species in this area are also deeply divergent even within a single river system. A mitochondrial phylogeny revealed that a ricefish population newly discovered from Cerekang River is sister to Oryzias dopingdopingensis Mandagi, Mokodongan, Tanaka, & Yamahira, another riverine species endemic to Doping-doping River. However, the Cerekang Oryzias was genetically isolated from O. dopingdopingensis , despite that Cerekang River and Doping-doping River share a connection across estuarine waters. This separation was supported by phylogenomic trees and population genetic structure analyses based on genome-wide single nucleotide polymorphisms. Coalescent-based demographic inference demonstrated that the ancestral population of these two riverine ricefishes had experienced a substantial population decrease and subsequently diverged into two sub-populations. Because the Cerekang Oryzias was also morphologically distinguished from O. dopingdopingensis , we described it as a new species, O. landangiensis. We infer that O. landangiensis and O. dopingdopingensis are of lake-origin and are relic species which were left in these rivers after the lake disappeared, and that they have lost their dispersal ability when inhabiting the ancient lake. The lost dispersal ability possibly contributed to the formation of the biodiversity hotspot for this fish group on this island. [ABSTRACT FROM AUTHOR]