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

Lorenzo Brandolese; Yves Meyer

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 273-285
  • ISSN: 1292-8119

Abstract

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We consider the spatial behavior of the velocity field u(x, t) of a fluid filling the whole space n ( n ≥ 2 ) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions ∫ u h ( x , t ) u k ( x , t ) d x = c ( t ) δ 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.

How to cite

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Brandolese, Lorenzo, and Meyer, Yves. "On the Instantaneous Spreading for the Navier–Stokes System in the Whole Space." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 273-285. <http://eudml.org/doc/90649>.

@article{Brandolese2010,
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. },
author = {Brandolese, Lorenzo, Meyer, Yves},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Navier–Stokes equations; space-decay; symmetries.; Navier-Stokes equations; symmetries},
language = {eng},
month = {3},
pages = {273-285},
publisher = {EDP Sciences},
title = {On the Instantaneous Spreading for the Navier–Stokes System in the Whole Space},
url = {http://eudml.org/doc/90649},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Brandolese, Lorenzo
AU - Meyer, Yves
TI - On the Instantaneous Spreading for the Navier–Stokes System in the Whole Space
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 273
EP - 285
AB - 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.
LA - eng
KW - Navier–Stokes equations; space-decay; symmetries.; Navier-Stokes equations; symmetries
UR - http://eudml.org/doc/90649
ER -

References

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  1. L. Brandolese, On the Localization of Symmetric and Asymmetric Solutions of the Navier-Stokes Equations dans n . C. R. Acad. Sci. Paris Sér. I Math332 (2001) 125-130.  
  2. Y. Dobrokhotov and A.I. Shafarevich, 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.  
  3. T. Gallay and C.E. Wayne, Long-time asymptotics of the Navier-Stokes and vorticity equations on 3 . Preprint. Univ. Orsay (2001).  
  4. C. He and Z. Xin, On the decay properties of Solutions to the nonstationary Navier-Stokes Equations in 3 . Proc. Roy. Soc. Edinburgh Sect. A131 (2001) 597-619.  
  5. T. Kato, Strong Lp-Solutions of the Navier-Stokes Equations in m , with applications to weak solutions. Math. Z.187 (1984) 471-480.  
  6. O. Ladyzenskaija, The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, English translation, Second Edition (1969).  
  7. T. Miyakawa, On space time decay properties of nonstationary incompressible Navier-Stokes flows in n . Funkcial. Ekvac.32 (2000) 541-557.  
  8. S. Takahashi, A wheighted equation approach to decay rate estimates for the Navier-Stokes equations. Nonlinear Anal.37 (1999) 751-789.  

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