Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces

When estimating solutions of dissipative partial differential equations in L-p-related spaces, we often need lower bounds for an integral involving the dissipative term. If the dissipative term is given by the usual Laplacian -Delta, lower bounds can be derived through integration by parts and embedding inequalities. However, when the Laplacian is replaced by the fractional Laplacian (-Delta)(alpha), the approach of integration by parts no longer applies. In this paper, we obtain lower bounds for the integral involving (-Delta)(alpha) by combining pointwise inequalities for (-Delta)(alpha) with Bernstein's inequalities for fractional derivatives. As an application of these lower bounds, we establish the existence and uniqueness of solutions to the generalized Navier-Stokes equations in Besov spaces. The generalized Navier-Stokes equations are the equations resulting from replacing -Delta in the Navier-Stokes equations by (-Delta)(alpha).