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Large-frequency global regularity for the incompressible Navier–Stokes equation

Published online by Cambridge University Press:  14 November 2011

Joel D. Avrin
Joel D. Avrin
Affiliation:
Department of Mathematics, University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, NC 28223-0001, USA
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Abstract

We obtain global existence and regularity of strong solutions to the incompressible Navier–Stokes equations for a variety of boundary conditions in such a way that the initial and forcing data can be large in the high-frequency eigenspaces of the Stokes operator. We do not require that the domain be thin as in previous analyses. But in the case of thin domains (and zero Dirichlet boundary conditions) our results represent a further improvement and refinement of previous results obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 5 , 1999 , pp. 903 - 912
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Avrin, J. D.. Large-eigenvalue global existence and regularity results for the Navier–Stokes equation. J. Diff. Eqns 127 (1996), 365–390.CrossRefGoogle Scholar
2Avrin, J. D.. A one-point attractor theory for the Navier–Stokes equation on thin domains with no-slip boundary conditions. Proc. AMS 127 (1999), 725–735.CrossRefGoogle Scholar
3Foias, C. and Prodi, G.. Sur le comportement global des solutions non stationaires des équations de Navier–Stokes en dimension 2. Rend. Sem. Mat. Univ. Padova 39 (1967), 1–34.Google Scholar
4Friedman, A.. Partial differential equations (New York: Holt, Rinehart and Winston, 1969).Google Scholar
5Fujita, H. and Kato, T.. On the Navier–Stokes initial value problem I. Arch. Ration. Mech. Analysis 16 (1964), 269–315.CrossRefGoogle Scholar
6Giga, Y. and Miyakawa, T.. Solutions in L r of the Navier–Stokes initial value problem. Arch. Ration. Mech. Analysis 89 (1985), 267–281.CrossRefGoogle Scholar
7Guillopé, C.. Comportement è l'infini des solutions des équations de Navier–Stokes et propriété des ensembles fonctionnels invariants (au attracteurs). Ann. Inst. Fourier (Grenoble) 32 (1982), 1–37.CrossRefGoogle Scholar
8Heywood, J. G.. The Navier–Stokes equations: on the existence, regularity and decay of solutions. Ind. U. Math. J. 29 (1980), 639–681.CrossRefGoogle Scholar
9Hopf, E.. Über die Anfangswertaufgabe für die hydrodynamischen Grudgleichungen. Math. Nachr. 4 (1951), 213–231.CrossRefGoogle Scholar
10Ladyzhenskaya, O. A.. The mathematical theory of viscous incompressible flow, 2nd edn (English transl.) (New York: Gordon and Breach, 1969).Google Scholar
11Leray, J.. Etude de diverses équations intégrales nonlinéaires et de quelques problémes que pose l'hydrodynamique. J. Math. Pures Appl. 12 (1933), 1–82.Google Scholar
12Raugel, G. and Sell, G. R.. Navier–Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions. J. Am. Math. Soc. 6 (1993), 503–568.Google Scholar
13Serrin, J.. On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Analysis 9 (1962), 187–195.CrossRefGoogle Scholar
14Témam, R.. Navier–Stokes equations (Amsterdam: North-Holland, 1977).Google Scholar
15Témam, R.. Navier–Stokes equations and nonlinear functional analysis. CBMS Regional Conference Series, no. 41 (Philadelphia: SIAM, 1983).Google Scholar