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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A one-point attractor theory for the Navier-Stokes equation on thin domains with no-slip boundary conditions
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by Joel D. Avrin PDF
Proc. Amer. Math. Soc. 127 (1999), 725-735 Request permission

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
  • Email: jdavrin@email.uncc.edu
  • Received by editor(s): August 7, 1996
  • Received by editor(s) in revised form: June 9, 1997
  • Communicated by: Jeffrey B. Rauch
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 725-735
  • MSC (1991): Primary 35B40, 35Q10
  • DOI: https://doi.org/10.1090/S0002-9939-99-04864-9
  • MathSciNet review: 1605915