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Asymptotic behavior of solutions to the three-dimensional Navier-Stokes equations. (English) Zbl 0759.35036
Summary: We study the lower bounds of rates of decay to solutions of the incompressible Navier-Stokes equations in three spatial dimensions. We show that if the initial data is solenoidal, has zero average and belongs to \(L^ 2\cap L^ 1\cap M^ c\), where \(M\) is the set of functions with radially equidistributed energy, then the corresponding solution \(u(x,t)\) to the Navier-Stokes equations has the following lower and upper bounds of decay: \[ C_ 0(t+1)^{-n/2-1} \leq \| u(\cdot,t)\|_{L^ 2}^ 2 \leq C_ 1(t+1)^{-n/2-1}. \]

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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