## Advances in Differential Equations

- Adv. Differential Equations
- Volume 12, Number 7 (2007), 721-736.

### Global solvability of the Navier-Stokes equations in spaces based on sum-closed frequency sets

Yoshikazu Giga, Katsuya Inui, Alex Mahalov, and Jürgen Saal

#### Abstract

We prove existence of global regular solutions for the $3$D Navier-Stokes equations with (or without) Coriolis force for a class of initial data $u_0$ in the space ${{{\mathrm{FM}}}_{\sigma,\delta}}$, i.e., for functions whose Fourier image ${\widehat{u}}_0$ is a vector-valued Radon measure and that are supported in sum-closed frequency sets with distance $\delta$ from the origin. In our main result we establish an upper bound for admissible initial data in terms of the Reynolds number, uniform on the Coriolis parameter $\Omega$. In particular this means that this upper bound is linearly growing in $\delta$. This implies that we obtain global-in-time regular solutions for large (in norm) initial data $u_0$ which may not decay at space infinity, provided that the distance $\delta$ of the sum-closed frequency set from the origin is sufficiently large.

#### Article information

**Source**

Adv. Differential Equations, Volume 12, Number 7 (2007), 721-736.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867432

**Mathematical Reviews number (MathSciNet)**

MR2331521

**Zentralblatt MATH identifier**

1144.76009

**Subjects**

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Secondary: 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76D05: Navier-Stokes equations [See also 35Q30] 76U05: Rotating fluids

#### Citation

Giga, Yoshikazu; Inui, Katsuya; Mahalov, Alex; Saal, Jürgen. Global solvability of the Navier-Stokes equations in spaces based on sum-closed frequency sets. Adv. Differential Equations 12 (2007), no. 7, 721--736. https://projecteuclid.org/euclid.ade/1355867432