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Fractional Navier-Stokes equations

  • * Corresponding author:Jan W. Cholewa

    * Corresponding author:Jan W. Cholewa 

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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  • 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.

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


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