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Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces. (English) Zbl 1104.35037
Summary: 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 \((-\Delta)^{\alpha}\).

MSC:
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
26A33 Fractional derivatives and integrals
76D05 Navier-Stokes equations for incompressible viscous fluids
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
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