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The H1-compact global attractor for the solutions to the Navier-Stokes equations in two-dimensional unbounded domains

Ju, Ning (2000) The H1-compact global attractor for the solutions to the Navier-Stokes equations in two-dimensional unbounded domains. Nonlinearity, 13 (4). pp. 1227-1238. ISSN 0951-7715. doi:10.1088/0951-7715/13/4/313.

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We extend previous results obtained by Rosa (1998 Nonlinear Anal. 32 71-85) on the existence of the global attractor for the two-dimensional Navier-Stokes equations on some unbounded domains. We show that if the forcing term is in the natural space H, then the global attractor is compact not only in the L2 norm but also in the H1 norm, and it attracts all bounded sets in H in the metric of V. The proof is based on the concept of asymptotic compactness and the use of the enstrophy equation. As compared with the work of Rosa, which proved the compactness and the attraction in the L2 norm, the new difficulty comes from the fact that the nonlinear term of the Navier-Stokes equations does not disappear from the enstrophy equation, while it does disappear in the energy equation due to its antisymmetry property.

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Additional Information:© Institute of Physics and IOP Publishing Limited 2000. Received 20 October 1999, in final form 21 March 2000; Print publication: Issue 4 (July 2000) This work was partially supported by the Department of Mathematical Sciences of the National Science Foundation under the grant NSF-DMS 9705229 and by the Research Fund of Indiana University. The author wishes to thank Professor Roger Temam for his constant support and encouragement. The author also wishes to thank the referees for their careful reading and helpful comments which have improved the original version of the manuscript.
Issue or Number:4
Record Number:CaltechAUTHORS:JUNnonlin00b
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4685
Deposited By: Archive Administrator
Deposited On:03 Sep 2006
Last Modified:08 Nov 2021 20:19

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