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.
Citation
Yasunori Maekawa. Yutaka Terasawa. "The Navier-Stokes equations with initial data in uniformly local $L^p$ spaces." Differential Integral Equations 19 (4) 369 - 400, 2006. https://doi.org/10.57262/die/1356050505
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