On the instantaneous spreading for the Navier-Stokes system in the whole space

Brandolese, Lorenzo; Meyer, Yves

doi:10.1051/cocv:2002021

Brandolese, Lorenzo ; Meyer, Yves

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 273-285.

Résumé

We consider the spatial behavior of the velocity field $u (x, t)$ of a fluid filling the whole space $ℝ^{n}$ ( $n \geq 2$ ) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions $\int u_{h} (x, t) u_{k} (x, t) d x = c (t) δ_{h, k}$ 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.

Zbl 1080.35063

DOI : https://doi.org/10.1051/cocv:2002021
Classification : 35B40, 76D05, 35Q30
Mots clés : 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},
     pages = {273--285},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     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, Tome 8 (2002) , pp. 273-285. doi : 10.1051/cocv:2002021. http://www.numdam.org/item/COCV_2002__8__273_0/

Bibliographie

[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.

[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