This paper by Tosio Kato focuses on the existence and decay properties of strong $L^p$-solutions to the Navier-Stokes equation in $\mathbb{R}^m$ for $m = 2, 3, \ldots$. The main results are presented in two groups: one assuming the initial velocity $a \in PL^m$ and the other assuming $a \in PL^m \cap PL^p$. Key theorems include: 1. **Theorem 1**: For $a \in PL^m$, there exists a unique solution $u$ such that $u$ and its derivatives decay exponentially as $t \to 0$ and $t \to \infty$. The solution also belongs to a specific space $E(0, T_1; PL^q)$ for $1/r = (1 - m/q)/2$ and $m < q < m^2/(m-2)$. 2. **Theorem 2**: If $\|a\|_m \leq \lambda$, the solution $u$ is global, and the norms $\|u(t)\|_q$ and $\|\partial u(t)\|_q$ decay exponentially as $t \to \infty$. 3. **Theorem 2'**: The integral $\int_0^T \|u(t)\|_m dt$ tends to zero as $T \to \infty$, indicating decay of the $L^m$ norm. 4. **Theorem 3**: For $a \in PL^m \cap PL^p$, the solution $u$ has additional decay properties, including $u$ and $t^{1/2} \partial u$ belonging to a specific space $BC([0, T_2]; PL^m \cap PL^p)$. 5. **Theorem 4**: If $\|a\|_m \leq \lambda_1$, the solution $u$ is global, and the norms $t^{(m(p-m/q)/2)u}$ and $t^{(m(p-m/q+1)/2) \partial u}$ decay exponentially as $t \to \infty$. These theorems provide a detailed analysis of the behavior of strong solutions to the Navier-Stokes equation in unbounded domains, addressing both local and global existence and decay properties.This paper by Tosio Kato focuses on the existence and decay properties of strong $L^p$-solutions to the Navier-Stokes equation in $\mathbb{R}^m$ for $m = 2, 3, \ldots$. The main results are presented in two groups: one assuming the initial velocity $a \in PL^m$ and the other assuming $a \in PL^m \cap PL^p$. Key theorems include: 1. **Theorem 1**: For $a \in PL^m$, there exists a unique solution $u$ such that $u$ and its derivatives decay exponentially as $t \to 0$ and $t \to \infty$. The solution also belongs to a specific space $E(0, T_1; PL^q)$ for $1/r = (1 - m/q)/2$ and $m < q < m^2/(m-2)$. 2. **Theorem 2**: If $\|a\|_m \leq \lambda$, the solution $u$ is global, and the norms $\|u(t)\|_q$ and $\|\partial u(t)\|_q$ decay exponentially as $t \to \infty$. 3. **Theorem 2'**: The integral $\int_0^T \|u(t)\|_m dt$ tends to zero as $T \to \infty$, indicating decay of the $L^m$ norm. 4. **Theorem 3**: For $a \in PL^m \cap PL^p$, the solution $u$ has additional decay properties, including $u$ and $t^{1/2} \partial u$ belonging to a specific space $BC([0, T_2]; PL^m \cap PL^p)$. 5. **Theorem 4**: If $\|a\|_m \leq \lambda_1$, the solution $u$ is global, and the norms $t^{(m(p-m/q)/2)u}$ and $t^{(m(p-m/q+1)/2) \partial u}$ decay exponentially as $t \to \infty$. These theorems provide a detailed analysis of the behavior of strong solutions to the Navier-Stokes equation in unbounded domains, addressing both local and global existence and decay properties.