# On the instantaneous spreading for the Navier–Stokes system in the whole space

Lorenzo Brandolese; Yves Meyer

ESAIM: Control, Optimisation and Calculus of Variations (2002)

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

## Access Full Article

top## Abstract

top## How to cite

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 (2002): 273-285. <http://eudml.org/doc/245397>.

@article{Brandolese2002,

abstract = {We consider the spatial behavior of the velocity field $u(x, t)$ of a fluid filling the whole space $\mathbb \{R\}^n$ ($n\ge 2$) 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},

language = {eng},

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/245397},

volume = {8},

year = {2002},

}

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

PY - 2002

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 $\mathbb {R}^n$ ($n\ge 2$) 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

UR - http://eudml.org/doc/245397

ER -

## References

top- [1] L. Brandolese, On the Localization of Symmetric and Asymmetric Solutions of the Navier–Stokes Equations dans ${\mathbb{R}}^{n}$. C. R. Acad. Sci. Paris Sér. I Math 332 (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 ${\mathbb{R}}^{3}$. Preprint. Univ. Orsay (2001).
- [4] C. He and Z. Xin, On the decay properties of Solutions to the nonstationary Navier–Stokes Equations in ${\mathbb{R}}^{3}$. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 597-619. Zbl0982.35083
- [5] T. Kato, Strong ${L}^{p}$-Solutions of the Navier–Stokes Equations in ${\mathbb{R}}^{m}$, 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). Zbl0184.52603MR254401
- [7] T. Miyakawa, On space time decay properties of nonstationary incompressible Navier–Stokes flows in ${\mathbb{R}}^{n}$. 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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.