On Solutions with Fast Decay of Nonstationary Navier—Stokes System in the Half-Space

2002 ◽  
pp. 91-120 ◽  
Author(s):  
Yoshiko Fujigaki ◽  
Tetsuro Miyakawa
Keyword(s):  
Half Space ◽  
Stokes System ◽  
Navier Stokes ◽  
Fast Decay

Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants

2016 ◽  
Vol 14 (05) ◽  
pp. 679-703 ◽  
Author(s):  
Renjin Jiang ◽  
Jie Xiao ◽  
Dachun Yang
Keyword(s):  
Heat Equation ◽  
Half Space ◽  
Harmonic Functions ◽  
Stokes System ◽  
Navier Stokes ◽  
Campanato Space ◽  
Poisson Integrals ◽  
Scaling Invariant

For [Formula: see text] and [Formula: see text], let [Formula: see text] be the space of harmonic functions [Formula: see text] on the upper half-space [Formula: see text] satisfying [Formula: see text] and [Formula: see text] be the Campanato space on [Formula: see text]. We show that [Formula: see text] coincides with [Formula: see text] for all [Formula: see text], where the case [Formula: see text] was originally discovered by Fabes, Johnson and Neri [E. B. Fabes, R. L. Johnson and U. Neri, Spaces of harmonic functions representable by Poisson integrals of functions in BMO and [Formula: see text], Indiana Univ. Math. J. 25 (1976) 159–170] and yet the case [Formula: see text] was left open. Moreover, for the scaling invariant version of [Formula: see text], [Formula: see text], which comprises all harmonic functions [Formula: see text] on [Formula: see text] satisfying [Formula: see text] we show that [Formula: see text], where [Formula: see text] is the collection of all functions [Formula: see text] such that [Formula: see text] are in [Formula: see text]. Analogues for solutions to the heat equation are also established. As an application, we show that the spaces [Formula: see text] unify naturally [Formula: see text], [Formula: see text] and [Formula: see text] which can be effectively adapted and applicable to suit handling the well/ill-posedness of the incompressible Navier–Stokes system on [Formula: see text].

scholarly journals A necessary condition of potential blowup for the Navier–Stokes system in half-space

2016 ◽  
Vol 369 (3-4) ◽  
pp. 1327-1352 ◽  
Author(s):  
T. Barker ◽  
G. Seregin
Keyword(s):  
Half Space ◽  
Stokes System ◽  
Navier Stokes ◽  
Necessary Condition

scholarly journals Green tensor of the Stokes system and asymptotics of stationary Navier–Stokes flows in the half space

2018 ◽  
Vol 323 ◽  
pp. 326-366 ◽  
Author(s):  
Kyungkeun Kang ◽  
Hideyuki Miura ◽  
Tai-Peng Tsai
Keyword(s):  
Half Space ◽  
Stokes System ◽  
Navier Stokes ◽  
Green Tensor ◽  
Stokes Flows

scholarly journals Some preliminary observations on a defect Navier–Stokes system

2019 ◽  
Vol 347 (10) ◽  
pp. 677-684 ◽  
Author(s):  
Amit Acharya ◽  
Roger Fosdick
Keyword(s):  
Stokes System ◽  
Navier Stokes

scholarly journals Asymptotic analysis of an optimal control problem for a viscous incompressible fluid with Navier slip boundary conditions

2021 ◽  
pp. 1-21
Author(s):  
Claudia Gariboldi ◽  
Takéo Takahashi
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Boundary Conditions ◽  
Control Problem ◽  
Stokes System ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Slip ◽  
Slip Boundary ◽  
Navier Slip Boundary Conditions

We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by α the friction coefficient and we analyze the asymptotic behavior of such a problem as α → ∞. More precisely, we prove that if we take an optimal control for each α, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.

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