The Navier–Stokes equations with body forces decaying coherently in time

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The long-time behavior of solutions of the three-dimensional Navier–Stokes equations in a periodic domain is studied. The time-dependent body force decays, as time t tends to infinity, in a coherent manner. In fact, it is assumed to have a general and complicated asymptotic expansion which involves complex powers of et, t, ln⁡t, or other iterated logarithmic functions of t. We prove that all Leray–Hopf weak solutions admit an asymptotic expansion which is independent of the solutions and is uniquely determined by the asymptotic expansion of the body force. The proof makes use of the complexifications of the Gevrey–Sobolev spaces together with those of the Stokes operator and the bilinear form of the Navier–Stokes equations.

Original languageEnglish
Article number127863
JournalJournal of Mathematical Analysis and Applications
Issue number2
StatePublished - Mar 15 2024


  • Asymptotic expansions
  • Complicated expansions
  • Fluid dynamics
  • Long-time behavior
  • Navier–Stokes equations


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