Stabilization of the chemotaxis–Navier–Stokes equations: Maximal regularity approach

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Abstract

Consider the chemotaxis–Navier–Stokes equations in a bounded smooth domain Ω⊂Rd for d≥3. We show that any solution starting close to an equilibrium exists globally and converges exponentially fast to the equilibrium as time tends to infinity, provided that the initial density n0 of amoebae satisfies ∫Ωn0dx<2|Ω|, where |Ω| stands for the Lebesgue measure of Ω. First, we prove the existence of a local strong solution for large initial data. Then, the global existence result is obtained assuming that the initial data are close to the equilibrium in their natural norm. In particular, we show the strong solution in the maximal Lp−Lq-regularity class with (p,q)∈(2,∞)×(d,∞) satisfying 2/p+d/q<1. Furthermore, the solution is real analytic in space and time.

Original languageEnglish
Article number125422
JournalJournal of Mathematical Analysis and Applications
Volume504
Issue number2
DOIs
Publication statusPublished - 2021 Dec 15

Keywords

  • Chemotaxis–Navier–Stokes equations
  • Maximal regularity
  • Stabilization
  • Well-posedness

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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