Abstract
We consider fractional Navier-Stokes equations in a smooth bound-ed domain \begin{document} $Ω\subset\mathbb{R}^N$ \end{document} , \begin{document} $N≥2$ \end{document} . 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.
Highlights
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. ∗ Corresponding author: Jan W
Adapting to the case of bounded domain formulation of an open problem in [11, (A), p. 2] we prove existence of Leray-Hopf type solution satisfying
We visualize the dependence of the parabolic regularization effect for weak solutions on [L2(Ω)]N -norm of initial data which all together lead to the statements as in Theorem 1.2. (iii) For the 3-D homogeneous Navier-Stokes equation, proving existential alternative, we obtain an estimate of the time in which the strong solution may cease to exist
Summary
We emphasize that the result like the one in Theorem 1.1 is obtained in the main body of the paper even for non-Hilbert phase spaces via regularization of the mild γ-solution, which we study in detail following Definition 2.1. (i) For the fractional power problems we get global existence of the 3-D regular solutions to (1.3) in a borderline case of parameter. (ii) We construct Leray’s type global weak solutions of the N -D Navier-Stokes equations which simultaneously satisfy Duhamel’s formula (see Lemmas 3.3, 3.4 and Theorem 3.6). (iii) For the 3-D homogeneous Navier-Stokes equation, proving existential alternative (see Theorem 1.3), we obtain an estimate of the time in which the strong solution may cease to exist.
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