New Asymptotic Profiles of Nonstationnary Solutions of the Navier-Stokes System

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dc.creatorBrandolese, Lorenzo
dc.creatorVigneron, Francois
dc.date2007-06-11
dc.date.accessioned2026-07-07T08:04:56Z
dc.date.available2026-07-07T08:04:56Z
dc.descriptionWe show that solutions $u(x,t)$ of the non-stationnary incompressible Navier--Stokes system in $\R^d$ ($d\geq2$) starting from mild decaying data $a$ behave as $|x|\to\infty$ as a potential field: u(x,t) = e^{tΔ}a(x) + γ_d\nabla_x(\sum_{h,k} \frac{δ_{h,k}|x|^2 - d x_h x_k}{d|x|^{d+2}} K_{h,k}(t))+\mathfrak{o}(\frac{1}{|x|^{d+1}}) where $γ_d$ is a constant and $K_{h,k}=\int_0^t(u_h| u_k)_{L^2}$ is the energy matrix of the flow. We deduce that, for well localized data, and for small $t$ and large enough $|x|$, c t |x|^{-(d+1)} \le |u(x,t)|\le c' t |x|^{-(d+1)}, where the lower bound holds on the complementary of a set of directions, of arbitrary small measure on $\mathbb{S}^{d-1}$. We also obtain new lower bounds for the large time decay of the weighted-$L^p$ norms, extending previous results of Schonbek, Miyakawa, Bae and Jin.
dc.description26 pages, article to appear in Journal de Mathématiques Pures et Appliquées
dc.identifierhttps://arxiv.org/abs/0706.1489
dc.identifierhttp://arxiv.org/abs/0706.1489
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/130143
dc.subjectAnalysis of PDEs
dc.subjectMathematical Physics
dc.subject76D05; 35Q30
dc.titleNew Asymptotic Profiles of Nonstationnary Solutions of the Navier-Stokes System
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