The Navier-Stokes equations with initial data in uniformly local $L^p$ spaces

In this paper we will construct local mild solutions of the Cauchy problem for the incompressible homogeneous Navier-Stokes equations in $d$-dimensional Euclidian space with initial data in uniformly local $ L^{p} $ ($ L^{p}_{uloc}$) spaces where $ p $ is greater than or equal to $d$. For the proof, we shall establish $L^p_{uloc}-L^q_{uloc}$ estimates for some convolution operators. We will also show that the mild solution associated with $ L^{d}_{uloc} $ almost periodic initial data at time zero becomes uniformly local almost periodic ($L^{\infty}$-almost periodic ) in any positive time.