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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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