Schonbek, Maria E. Asymptotic behavior of solutions to the three-dimensional Navier-Stokes equations. (English) Zbl 0759.35036 Indiana Univ. Math. J. 41, No. 3, 809-823 (1992). 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: \[ C_ 0(t+1)^{-n/2-1} \leq \| u(\cdot,t)\|_{L^ 2}^ 2 \leq C_ 1(t+1)^{-n/2-1}. \] Cited in 1 ReviewCited in 58 Documents MSC: 35Q30 Navier-Stokes equations 35B40 Asymptotic behavior of solutions to PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids Keywords:lower bounds of rates of decay; incompressible Navier-Stokes equations; initial data PDFBibTeX XMLCite \textit{M. E. Schonbek}, Indiana Univ. Math. J. 41, No. 3, 809--823 (1992; Zbl 0759.35036) Full Text: DOI