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(x,t) of a fluid filling the whole space n (n2) 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)dx=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.

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 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 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. A 131 (2001) 597-619. | Zbl 0982.35083

[5] T. Kato, Strong L p -Solutions of the Navier-Stokes Equations in 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 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