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A one-point attractor theory for the
Navier-Stokes equation on thin domains
with no-slip boundary conditions

Author: Joel D. Avrin
Journal: Proc. Amer. Math. Soc. 127 (1999), 725-735
MSC (1991): Primary 35B40, 35Q10
MathSciNet review: 1605915
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Abstract: In an earlier paper related to recent results of Raugel and Sell for periodic boundary conditions, we considered the incompressible Navier-Stokes equations on 3-dimensional thin domains with zero (``no-slip'') boundary conditions and established global regularity results. We extend those results here by developing an attractor theory. We first show that under similar thinness restrictions trajectories of solutions approach each other in $L^4$-norm exponentially. Next, for constant-in-time forcing data $ f_1=f_1\left( x\right) ,$ we suppose that $f\left( t\right) \rightarrow f_1$ in $L^2$ as $t\rightarrow +\infty ,$ and show that if $v$ and $w_1$ solve the equations with forcing data $f$ and $f_1$, respectively, then $\left\| v\left( t\right) -w_1\left( t\right) \right\| _4\rightarrow 0$ as $ t\rightarrow +\infty .$ For similar thinness restrictions we show that the steady-flow equations with forcing data $f_1$ have a unique solution $u_s$. Under both thinness assumptions we then have that all solutions $v\left( t\right) $ converge to $ u_s$ in $L_4$ as $t\rightarrow +\infty $; thus we have a one-point attractor for strong solutions. In fact, we have a one-point attractor for the Leray solutions as well. Moreover, under significantly more relaxed thinness assumptions we are able to show that Leray solutions nonetheless eventually become regular.

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Additional Information

Joel D. Avrin
Affiliation: Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223

Received by editor(s): August 7, 1996
Received by editor(s) in revised form: June 9, 1997
Communicated by: Jeffrey B. Rauch
Article copyright: © Copyright 1999 American Mathematical Society