ESAIM: Control, Optimisation and Calculus of Variations

Research Article

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

Brandolese, Lorenzoa1a2 and Meyer, Yvesa3a2

a1 Centre de Mathématiques et de leurs Applications, ENS de Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex, France; brandole@cmla.ens-cachan.fr.

a2 Équipe Modal'X, bâtiment G, Université de Paris X – Nanterre, 200 avenue de la République, 92001 Nanterre Cedex, France.

a3 Centre de Mathématiques et de leurs Applications, ENS de Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex, France; ymeyer@cmla.ens-cachan.fr.

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.

(Received November 23 2001)

(Revised December 13 2001)

(Online publication August 15 2002)

Key Words:

  • Navier–Stokes equations;
  • space-decay;
  • symmetries.

Mathematics Subject Classification:

  • 35B40;
  • 76D05;
  • 35Q30
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