Analysis & PDE

Global regularity for the Navier–Stokes equations with some classes of large initial data

Marius Paicu and Zhifei Zhang

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Abstract

Chemin, Gallagher, and Paicu obtained in 2010 a class of large initial data that generate a global smooth solution to the three-dimensional, incompressible Navier–Stokes equation. The data varies slowly in the vertical direction — it is expressed as a function of εx3 — and it has a norm that blows up as the small parameter goes to zero. This type of initial data can be regarded as an ill prepared case, in contrast with the well prepared case treated in earlier papers. The data was supposed to evolve in a special domain, namely Ω=Th2×v. The choice of a periodic domain in the horizontal variable played an important role.

The aim of this article is to study the case where the fluid evolves in the whole space 3. In this case, we have to overcome the difficulties coming from very low horizontal frequencies. We consider in this paper an intermediate situation between the well prepared case and ill prepared situation (the norms of the horizontal components of initial data are small but the norm of the vertical component blows up as the small parameter goes to zero). The proof uses the analytical-type estimates and the special structure of the nonlinear term of the equation.

Article information

Source
Anal. PDE, Volume 4, Number 1 (2011), 95-113.

Dates
Received: 27 March 2009
Revised: 14 May 2010
Accepted: 1 September 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731112

Digital Object Identifier
doi:10.2140/apde.2011.4.95

Mathematical Reviews number (MathSciNet)
MR2783307

Zentralblatt MATH identifier
1242.35187

Subjects
Primary: 35B65: Smoothness and regularity of solutions 35Q35: PDEs in connection with fluid mechanics 76D99: None of the above, but in this section 76N10: Existence, uniqueness, and regularity theory [See also 35L60, 35L65, 35Q30]

Keywords
Navier–Stokes equations global well-posedness large data

Citation

Paicu, Marius; Zhang, Zhifei. Global regularity for the Navier–Stokes equations with some classes of large initial data. Anal. PDE 4 (2011), no. 1, 95--113. doi:10.2140/apde.2011.4.95. https://projecteuclid.org/euclid.apde/1513731112


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