TY - JOUR

T1 - Stabilization of the chemotaxis–Navier–Stokes equations

T2 - Maximal regularity approach

AU - Watanabe, Keiichi

N1 - Funding Information:
This research was partly supported by JSPS KAKENHI Grant Number 20K22311 and 21K13826 and Waseda University Grant for Special Research Projects.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/12/15

Y1 - 2021/12/15

N2 - 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.

AB - 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.

KW - Chemotaxis–Navier–Stokes equations

KW - Maximal regularity

KW - Stabilization

KW - Well-posedness

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U2 - 10.1016/j.jmaa.2021.125422

DO - 10.1016/j.jmaa.2021.125422

M3 - Article

AN - SCOPUS:85107681358

VL - 504

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

M1 - 125422

ER -