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On the Instantaneous Spreading for the Navier–Stokes System in the Whole Space

Published online by Cambridge University Press:  15 August 2002

Lorenzo Brandolese  and
Yves Meyer
Lorenzo Brandolese
Affiliation:
Centre de Mathématiques et de leurs Applications, ENS de Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex, France; [email protected]. Équipe Modal'X, bâtiment G, Université de Paris X – Nanterre, 200 avenue de la République, 92001 Nanterre Cedex, France.
Yves Meyer
Affiliation:
Équipe Modal'X, bâtiment G, Université de Paris X – Nanterre, 200 avenue de la République, 92001 Nanterre Cedex, France. Centre de Mathématiques et de leurs Applications, ENS de Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex, France; [email protected].
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Abstract

We consider the spatial behavior of the velocity field u(x, t) of a fluid filling the whole space $\xR^n$ ($n\ge2$) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions $\int u_h(x,t)u_k(x,t)\,{\rm d}x=c(t)\delta_{h,k}$ under more general assumptions on the localization of u. We also give some new examples of solutions which have a stronger spatial localization than in the generic case.

Keywords

Navier–Stokes equationsspace-decaysymmetries.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 273 - 285
Copyright
© EDP Sciences, SMAI, 2002

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References

Brandolese, L., On the Localization of Symmetric and Asymmetric Solutions of the Navier-Stokes Equations dans $\xR^n$ . C. R. Acad. Sci. Paris Sér. I Math 332 (2001) 125-130. CrossRef
Dobrokhotov, Y. and Shafarevich, A.I., Some integral identities and remarks on the decay at infinity of solutions of the Navier-Stokes Equations. Russian J. Math. Phys. 2 (1994) 133-135.
T. Gallay and C.E. Wayne, Long-time asymptotics of the Navier-Stokes and vorticity equations on $\xR^3$ . Preprint. Univ. Orsay (2001).
He, C. and Xin, Z., On the decay properties of Solutions to the nonstationary Navier-Stokes Equations in $\xR^3$ . Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 597-619. CrossRef
Kato, T., Strong L p -Solutions of the Navier-Stokes Equations in $\xR^m$ , with applications to weak solutions. Math. Z. 187 (1984) 471-480. CrossRef
O. Ladyzenskaija, The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, English translation, Second Edition (1969).
Miyakawa, T., On space time decay properties of nonstationary incompressible Navier-Stokes flows in $\xR^n$ . Funkcial. Ekvac. 32 (2000) 541-557.
Takahashi, S., A wheighted equation approach to decay rate estimates for the Navier-Stokes equations. Nonlinear Anal. 37 (1999) 751-789. CrossRef