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.  Zbl0973.35149
  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.  Zbl0976.35508
  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.  Zbl0982.35083
  5. T. Kato, Strong Lp-Solutions of the Navier-Stokes Equations in m , with applications to weak solutions. Math. Z.187 (1984) 471-480.  Zbl0545.35073
  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.  Zbl1142.35545
  8. S. Takahashi, A wheighted equation approach to decay rate estimates for the Navier-Stokes equations. Nonlinear Anal.37 (1999) 751-789.  Zbl0941.35066

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