Differential and Integral Equations

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

Yasunori Maekawa and Yutaka Terasawa

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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

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

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 35B15: Almost and pseudo-almost periodic solutions 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]


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

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