The Generalized Incompressible Navier-Stokes Equations in Besov Spaces

This paper is concerned with global solutions of the generalized Navier-Stokes equations. The generalized Navier-Stokes equations here refer to the equations obtained by replacing the Laplacian in the Navier-Stokes equations by the more general operator (−∆) with α > 0. It has previously been shown that any classical solution of the d-dimensional generalized NavierStokes equations with α ≥ 1 2 + d 4 is always global in time. Thus, attention here is solely focused on the case when α < 1 2 + d 4 . We consider solutions emanating from initial data in several Besov spaces and establish the global existence and uniqueness of the solutions when the corresponding initial data are comparable to the diffusion coefficient in these Besov spaces.