# 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

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topBrandolese, 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

top- 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.
- 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.
- T. Gallay and C.E. Wayne, Long-time asymptotics of the Navier-Stokes and vorticity equations on ${}^{3}$. Preprint. Univ. Orsay (2001).
- 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.
- T. Kato, Strong Lp-Solutions of the Navier-Stokes Equations in ${}^{m}$, with applications to weak solutions. Math. Z.187 (1984) 471-480.
- O. Ladyzenskaija, The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, English translation, Second Edition (1969).
- T. Miyakawa, On space time decay properties of nonstationary incompressible Navier-Stokes flows in ${}^{n}$. Funkcial. Ekvac.32 (2000) 541-557.
- 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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