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

Autor: François Vigneron, Lorenzo Brandolese
Přispěvatelé: Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)
Jazyk: angličtina
Rok vydání: 2007
Předmět:
Mathematics(all)
General Mathematics
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]
FOS: Physical sciences
upper bound estimates
01 natural sciences
Measure (mathematics)
Upper and lower bounds
Mathematics - Analysis of PDEs
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]
far field asymptotic
lower bound estimates
FOS: Mathematics
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Far field asymptotics
Navier stokes
0101 mathematics
Mathematical Physics
Mathematics
vorticity
Asymptotic behavior at infinity
large distance behavior
Mild solutions to the Navier–Stokes system
76D05
35Q30
Applied Mathematics
010102 general mathematics
Mathematical analysis
35Q30
76D05
Peetre weight
Mathematical Physics (math-ph)
Stokes flow
010101 applied mathematics
Flow (mathematics)
Compressibility
Potential flow
Constant (mathematics)
Analysis of PDEs (math.AP)
Zdroj: Journal de Mathématiques Pures et Appliquées
Journal de Mathématiques Pures et Appliquées, Elsevier, 2007, ⟨10.1016/j.matpur.2007.04.007⟩
ISSN: 0021-7824
DOI: 10.1016/j.matpur.2007.04.007⟩
Popis: We 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^{t\Delta}a(x) + \gamma_d\nabla_x(\sum_{h,k} \frac{\delta_{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 $\gamma_d$ 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.
Comment: 26 pages, article to appear in Journal de Math\'ematiques Pures et Appliqu\'ees
Databáze: OpenAIRE
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