Analytic semigroup approach to generalized Navier-Stokes flows in Besov spaces

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 >= 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