Asymptotic expansions in a general system of decaying functions for solutions of the Navier–Stokes equations

Dat Cao, Luan Hoang

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Abstract

We study the long-time dynamics of the Navier–Stokes equations in the three-dimensional periodic domains with a body force decaying in time. We introduce appropriate systems of decaying functions and corresponding asymptotic expansions in those systems. We prove that if the force has a large-time asymptotic expansion in Gevrey–Sobolev spaces in such a general system, then any Leray–Hopf weak solution admits an asymptotic expansion of the same type. This expansion is uniquely determined by the force, and independent of the solutions. Various applications of the abstract results are provided which particularly include the previously obtained expansions for the solutions in case of power decay, as well as the new expansions in case of the logarithmic and iterated logarithmic decay.

Original languageEnglish
Pages (from-to)1023-1072
Number of pages50
JournalAnnali di Matematica Pura ed Applicata
Volume199
Issue number3
DOIs
StatePublished - Jun 1 2020

Keywords

  • Abstract expansions
  • Asymptotic expansions
  • Expansion theory
  • Long-time dynamics
  • Navier–Stokes equations

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