On the Navier-Stokes equations with initial data nondecaying at space infinity

Abstract

We will consider the Cauchy problem for the incompressible homogeneous Navier-Stokes equations in the $d$-dimensional Eucledian space with initial data in uniformly local $L^p\ (L_{uloc}^{p})$ spaces where $p$ is larger than or equal to $d$. For the construction of the local mild solution of this, $L_{uloc}^{p} - L_{uloc}^{q}$ estimates for some convolution operators are important. So we explain these estimates here.