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Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in \(\mathbb{R}^n\) and \(\mathbb{R}_+^n\). (English) Zbl 1012.35062
Salvi, Rodolfo (ed.), The Navier-Stokes equations: theory and numerical methods. Proceedings of the international conference, Varenna, Lecco, Italy, 2000. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 223, 205-219 (2002).
The author investigates the large-time profiles of weak and strong solutions of the Navier-Stokes equations in the whole space \(\mathbb{R}^n\) and in the half-space \(\mathbb{R}^n_+\), \(n\geq 2\), under some specific conditions on the initial velocity. The main results are the following: As \(t\to \infty\), the solutions in the whole space admit an asymptotic expansion in terms of the space-time derivatives of Gaussian-like functions, provided the initial velocity satisfies decay and moment conditions. In the case of the half-space, the asymptotic expansion involves only the normal derivatives of the mentioned functions. An application of these results to the analysis of the modes of energy decay in the half-space is given.
For the entire collection see [Zbl 0972.00046].
35Q30 Navier-Stokes equations
35C20 Asymptotic expansions of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids