Fractional Navier-Stokes equations

Jan W. Cholewa; Tomasz Dlotko

doi:10.3934/dcdsb.2017149

Article Contents

2018, Volume 23, Issue 8: 2967-2988. Doi: 10.3934/dcdsb.2017149

This issue Previous Article Positive solutions to the unstirred chemostat model with Crowley-Martin functional response Next Article Stability of dislocation networks of low angle grain boundaries using a continuum energy formulation

Fractional Navier-Stokes equations

Jan W. Cholewa^, and
Tomasz Dlotko

Institute of Mathematics, University of Silesia in Katowice, 40-007 Katowice, Poland

* Corresponding author:Jan W. Cholewa
* Corresponding author:Jan W. Cholewa

Received: December 2016

Revised: February 2017

Early access: March 2017

Published: September 2018

J.C. partially supported by grant MTM2012-31298 from Ministerio de Economia y Competividad, Spain; T.D. partially supported by NCN grant DEC-2012/05/B/ST1/00546, Poland

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Abstract

We consider fractional Navier-Stokes equations in a smooth bound-ed domain $Ω\subset\mathbb{R}^N$, $N≥2$. Following the geometric theory of abstract parabolic problems we give the detailed analysis concerning existence, uniqueness, regularization and continuation properties of the solution. For the original Navier-Stokes problem we construct next global solution of the Leray-Hopf type satisfying also Duhamel's integral formula. Focusing finally on the 3-D model with zero external force we estimate a time after which the latter solution regularizes to strong solution. We also estimate a time such that if a local strong solution exists until that time, then it exists for ever.

Keywords:

Navier-Stokes equations,
abstract parabolic equations,
initial-boundary value problems for pseudodifferential operators.

Mathematics Subject Classification: Primary: 35Q30, 35K90, 35S11.

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