This paper by Tosio Kato presents results on the existence and decay properties of strong $ L^p $-solutions to the Navier-Stokes equation in $ \mathbb{R}^m $, with applications to weak solutions. The Navier-Stokes equations are given as a system of partial differential equations describing the motion of fluid substances. The paper focuses on strong solutions $ u(t) \in PL^m $, which are solutions that exist locally in time if the initial velocity $ a $ is in $ PL^m $, and globally if $ \|a\|_m $ is sufficiently small. The paper addresses the lack of compactness in unbounded domains and the decay of global solutions as $ t \to \infty $, which is a nontrivial problem. The main results are summarized in several theorems. The first theorem states that if $ a \in PL^m $, then there exists a unique solution $ u $ such that $ t^{(1-m/q)/2}u \in BC([0,T); PL^q) $ for $ m \leq q \leq \infty $, and $ t^{1-m/2q}\hat{\partial}u \in BC([0,T); PL^q) $ for $ m \leq q < \infty $. The second theorem states that if $ \|a\|_m $ is sufficiently small, then the solution is global, and the $ L^q $-norm of $ u(t) $ decays like $ t^{-(1-m/q)/2} $ as $ t \to \infty $. The third theorem extends these results to the case where $ a \in PL^m \cap PL^p $, and the fourth theorem provides additional decay properties for global solutions. The paper also discusses the spatial decay of solutions and the behavior of solutions as $ t \to 0 $ and $ t \to \infty $. The results have implications for the study of weak solutions and the behavior of turbulent solutions.This paper by Tosio Kato presents results on the existence and decay properties of strong $ L^p $-solutions to the Navier-Stokes equation in $ \mathbb{R}^m $, with applications to weak solutions. The Navier-Stokes equations are given as a system of partial differential equations describing the motion of fluid substances. The paper focuses on strong solutions $ u(t) \in PL^m $, which are solutions that exist locally in time if the initial velocity $ a $ is in $ PL^m $, and globally if $ \|a\|_m $ is sufficiently small. The paper addresses the lack of compactness in unbounded domains and the decay of global solutions as $ t \to \infty $, which is a nontrivial problem. The main results are summarized in several theorems. The first theorem states that if $ a \in PL^m $, then there exists a unique solution $ u $ such that $ t^{(1-m/q)/2}u \in BC([0,T); PL^q) $ for $ m \leq q \leq \infty $, and $ t^{1-m/2q}\hat{\partial}u \in BC([0,T); PL^q) $ for $ m \leq q < \infty $. The second theorem states that if $ \|a\|_m $ is sufficiently small, then the solution is global, and the $ L^q $-norm of $ u(t) $ decays like $ t^{-(1-m/q)/2} $ as $ t \to \infty $. The third theorem extends these results to the case where $ a \in PL^m \cap PL^p $, and the fourth theorem provides additional decay properties for global solutions. The paper also discusses the spatial decay of solutions and the behavior of solutions as $ t \to 0 $ and $ t \to \infty $. The results have implications for the study of weak solutions and the behavior of turbulent solutions.