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Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces

Part of: Representations of solutions Partial differential equations Incompressible viscous fluids Equations of mathematical physics and other areas of application Infinite-dimensional dissipative dynamical systems

Published online by Cambridge University Press:  22 January 2019

Dat Cao  and
Luan Hoang
Dat Cao
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX79409-1042, USA (dat.cao@ttu.edu, luan.hoang@ttu.edu)
Luan Hoang
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX79409-1042, USA (dat.cao@ttu.edu, luan.hoang@ttu.edu)
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Abstract

The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

Keywords

Navier-Stokesasymptotic expansionpower-decaying

MSC classification

Primary: 35C20: Asymptotic expansions
Secondary: 35B40: Asymptotic behavior of solutions 35Q30: Navier-Stokes equations 37L99: None of the above, but in this section 76D05: Navier-Stokes equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 569 - 606
Copyright
Copyright © Royal Society of Edinburgh 2019

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