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

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.