## Differential and Integral Equations

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

#### Abstract

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.

#### Article information

Source
Differential Integral Equations, Volume 19, Number 4 (2006), 369-400.

Dates
First available in Project Euclid: 21 December 2012

Maekawa, Yasunori; Terasawa, Yutaka. The Navier-Stokes equations with initial data in uniformly local $L^p$ spaces. Differential Integral Equations 19 (2006), no. 4, 369--400. https://projecteuclid.org/euclid.die/1356050505