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\(L^ 2\) decay for weak solutions of the Navier-Stokes equations. (English) Zbl 0602.76031
This paper is concerned with the asymptotic behavior in time of solutions to the Navier-Stokes equations defined on the whole of three-dimensional space. In a classical paper of J. Leray [Acta. Math. 63, 193-248 (1934)] it was posed as a problem to determine whether or not weak solutions to the Navier-Stokes equations defined over all of three- dimensional space, decay.
This article answers this question and shows that if the externally applied force satisfies certain integrability and decay restrictions, then weak solutions do decay in \(L^ 2\)-norm. Furthermore, an upper bound for the decay rate is determined.
The paper first gives formal arguments to show that the \(L^ 2\)-norm of the solution decays at least as fast as \((1+t)^{-1/4}\) when the force is zero and as fast as \((1+t)^{-3/4}\) for a divergence free force which has decay behaviour at least as rapid. Rigorous arguments are then applied to ”Leray-Hopf” solutions constructed by L.Caffarelli, R. Kohn and L. Nirenberg [Commun. Pure Appl. Math. 35, 771-831 (1982; Zbl 0509.35067)]. In this case it is shown that the \(L^ 2({\mathbb{R}}^ 3)\) norm decays at least like \((1+t)^{-1/4}\), for suitable initial data, when f is again zero or divergence free with suitable asymptotic decay. It is observed that faster decay may be achieved if certain uniform boundedness conditions may be established.
Reviewer: B.Straughan

76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
Full Text: DOI
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