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

Published online by Cambridge University Press:  22 January 2019

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)
Corresponding

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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