On the instantaneous spreading for the Navier-Stokes system in the whole space
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8  (2002), p. 273-285

We consider the spatial behavior of the velocity field $u\left(x,t\right)$ of a fluid filling the whole space ${ℝ}^{n}$ ($n\ge 2$) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions $\int {u}_{h}\left(x,t\right){u}_{k}\left(x,t\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x=c\left(t\right){\delta }_{h,k}$ 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.

DOI : https://doi.org/10.1051/cocv:2002021
Classification:  35B40,  76D05,  35Q30
Keywords: Navier-Stokes equations, space-decay, symmetries
@article{COCV_2002__8__273_0,
author = {Brandolese, Lorenzo and Meyer, Yves},
title = {On the instantaneous spreading for the Navier-Stokes system in the whole space},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {8},
year = {2002},
pages = {273-285},
doi = {10.1051/cocv:2002021},
zbl = {1080.35063},
language = {en},
url = {http://www.numdam.org/item/COCV_2002__8__273_0}
}

Brandolese, Lorenzo; Meyer, Yves. On the instantaneous spreading for the Navier-Stokes system in the whole space. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002) , pp. 273-285. doi : 10.1051/cocv:2002021. http://www.numdam.org/item/COCV_2002__8__273_0/

[1] L. Brandolese, On the Localization of Symmetric and Asymmetric Solutions of the Navier-Stokes Equations dans ${ℝ}^{n}$. C. R. Acad. Sci. Paris Sér. I Math 332 (2001) 125-130. | Zbl 0973.35149

[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. | Zbl 0976.35508

[3] T. Gallay and C.E. Wayne, Long-time asymptotics of the Navier-Stokes and vorticity equations on ${ℝ}^{3}$. Preprint. Univ. Orsay (2001).

[4] C. He and Z. Xin, On the decay properties of Solutions to the nonstationary Navier-Stokes Equations in ${ℝ}^{3}$. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 597-619. | Zbl 0982.35083

[5] T. Kato, Strong ${L}^{p}$-Solutions of the Navier-Stokes Equations in ${ℝ}^{m}$, with applications to weak solutions. Math. Z. 187 (1984) 471-480. | Zbl 0545.35073

[6] O. Ladyzenskaija, The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, English translation, Second Edition (1969). | MR 254401 | Zbl 0184.52603

[7] T. Miyakawa, On space time decay properties of nonstationary incompressible Navier-Stokes flows in ${ℝ}^{n}$. Funkcial. Ekvac. 32 (2000) 541-557. | Zbl pre02112739

[8] S. Takahashi, A wheighted equation approach to decay rate estimates for the Navier-Stokes equations. Nonlinear Anal. 37 (1999) 751-789. | Zbl 0941.35066