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 Lp-related spaces, we often need lower bounds for an integral involving the dissipative term. If the dissipative term is given by the usual Laplacian −Δ, lower bounds can be derived through integration by parts and embedding inequalities. However, when the Laplacian is replaced by the fractional Laplacian (−Δ)α, the approach of integration by parts no longer applies. In this paper, we obtain lower bounds for the integral involving (−Δ)α by combining pointwise inequalities for (−Δ)α 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 −Δ in the Navier-Stokes equations by (−Δ)α.