Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in \(\mathbb{R}^n\)
Some estimations for the decay of higher-order derivatives of solutions to the unsteady unforced Navier-Stokes (N-S) equations in \(\mathbb{R}^n\) are obtained. The main result is an extension to \(\mathbb{R}^n\) of [\textit{C. Foias} and \textit{R. Temam}, J. Funct. Anal. 87, No. 2, 359-369 (1989; Zbl 0702.35203)], where periodic boundary conditions were considered. Some assumptions are used, concerning the algebraic decay of N-S solutions, but only for \(n\leq 3\). The main point is to prove that the solutions enter a subclass of real analytic functions in an arbitrary short time. For a large class of initial data, the N-S solution stays close to a solution of the heat equation in the \(L^2\) norm. By using this fact, lower bounds concerning the decay of derivatives can be obtained.
scientific article; zbMATH DE number 1443261
scientific article