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Journal of Mathematical Fluid Mechanics

, Volume 19, Issue 4, pp 709–724 | Cite as

Analytic Semigroup Approach to Generalized Navier–Stokes Flows in Besov Spaces

  • Zhi-Min ChenEmail author
Article
  • 204 Downloads

Abstract

The energy dissipation of the Navier–Stokes equations is controlled by the viscous force defined by the Laplacian \(-\Delta \), while that of the generalized Navier–Stokes equations is determined by the fractional Laplacian \((-\Delta )^\alpha \). The existence and uniqueness problem is always solvable in a strong dissipation situation in the sense of large \(\alpha \) but it becomes complicated when \(\alpha \) is decreasing. In this paper, the well-posedness regarding to the unique existence of small time solutions and small initial data solutions is examined in critical homogeneous Besov spaces for \(\alpha \ge 1/2\). An analytic semigroup approach to the understanding of the generalized Navier–Stokes equations is developed and thus the well-posedness on the equations is examined in a manner different to earlier investigations.

Keywords

Generalized Navier–Stokes equations well-posedness analytic semigroup Besov spaces 

Mathematics Subject Classification

35B32 35B35 35Q35 86A10 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Ship ScienceUniversity of SouthamptonSouthamptonUK
  2. 2.School of Mathematics and StatisticsShenzhen UniversityShenzhenChina

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