×

zbMATH — the first resource for mathematics

\(L^ 2\) 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

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Caffarelli, L.,Kohn, R. &Nirenberg, L., Partial regularity of suitably weak solutions of the Navier-Stokes equations, Comm. on Pure and Applied Math.25 (1982), 771-831. · Zbl 0509.35067 · doi:10.1002/cpa.3160350604
[2] Heywood, J., The Navier-Stokes equations: On the existence, regularity and decay of solutions, Indiana University Math. J.29 (1980), 639-681. · Zbl 0494.35077 · doi:10.1512/iumj.1980.29.29048
[3] Kato, T., On the Navier-Stokes equations, preprint.
[4] Kawashima, S.,Matsumura, A., &Nishida, T., On the fluiddynamical approximations to the Boltzmann equation at the level of the Navier-Stokes equation, Comm. Math. Phys.70 (1979), 97-124. · Zbl 0449.76053 · doi:10.1007/BF01982349
[5] Ladyzenskaya, O. A., The mathematical theory of viscous incompressible flows, 2nd ed. Gordon and Breach, New York, 1969.
[6] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math.63 (1934), 193-248. · JFM 60.0726.05 · doi:10.1007/BF02547354
[7] Masuda, K., On the stability of incompressible viscous fluid motions past objects, J. Math. Soc. Japan27 (1975), 294-327. · Zbl 0303.76011 · doi:10.2969/jmsj/02720294
[8] Schonbek, M., Uniform decay rates for parabolic conservation laws, Journal of Nonlinear Analysis: Theory, Methods and Applications, to appear. · Zbl 0617.35060
[9] Temam, R., Navier-Stokes equations. Theory and numerical analysis, North-Holland, Amsterdam and New York, 1979. · Zbl 0426.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.