Space–time regularity of the mild solutions to the incompressible generalized Navier–Stokes equations with small rough initial data. (April 2015)

• Record Type: Journal Article
• Authors: Liu, Qiao
• Is Part Of: Nonlinear analysis. Volume 22(2015)
• Page Start: 373
• Page End: 387
• Publication Date: 2015-04
• Subjects: Generalized Navier–Stokes equations -- Q-spaces -- Regularity -- Littlewood–Paley decomposition -- Besov space -- Trajectory; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2014.10.002
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In recent paper of Li and Zhai (2010), the authors proved the global-in-time existence and spatial regularity of mild solutions to the n -dimensional incompressible generalized Navier–Stokes equations with small initial data u 0 in Q α ; ∞ β, − 1 ( R n ) ≔ ∇ ⋅ ( Q α β ( R n ) ) n, β ∈ ( 1 2, 1 ] and α ∈ [ 0, β ) . In this paper, by using the Fourier localization method, we shall show that the mild solution presented by Li and Zhai (2010) satisfies the decay estimates for any space–time derivative involving some borderline Besov space norms. Moreover, the solution has a unique trajectory which is Hölder continuous with respect to space variables.