Blow-up prevention by quadratic degradation in a two-dimensional Keller–Segel–Navier–Stokes system

• Published in: Zeitschrift für angewandte Mathematik und Physik 2016-12, Vol.67 (6), p.1-23
• Main Authors: Tao, Youshan, Winkler, Michael
• Format: Article
• Language: English
• Subjects: Applications of mathematics; Boundary value problems; Degradation; Engineering; Fluid flow; Incompressible flow; Incompressible fluids; Initial value problems; Logistics; Mathematical analysis; Mathematical Methods in Physics; Microorganisms; Navier-Stokes equations; Theoretical and Applied Mechanics
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-016-0732-1

This paper deals with an initial-boundary value problem in a two-dimensional smoothly bounded domain for the Keller–Segel–Navier–Stokes system with logistic source, as given by n t + u · ∇ n = Δ n - ∇ · ( n ∇ c ) + r n - μ n 2 , c t + u · ∇ c = Δ c - c + n , u t + u · ∇ u = Δ u - ∇ P + n ∇ ϕ + g , ∇ · u = 0 , which describes the mutual interaction of chemotactically moving microorganisms and their surrounding incompressible fluid. It is shown that whenever μ > 0 , r ≥ 0 , g ∈ C 1 ( Ω ¯ × [ 0 , ∞ ) ) ∩ L ∞ ( Ω × ( 0 , ∞ ) ) and the initial data ( n 0 , c 0 , u 0 ) are sufficiently smooth fulfilling n 0 ≢ 0 , the considered problem possesses a global classical solution which is bounded. Moreover, if r = 0 , then this solution satisfies n ( · , t ) → 0 and c ( · , t ) → 0 in L ∞ ( Ω ) as t → ∞ , and if additionally ∫ 0 ∞ ∫ Ω | g ( x , t ) | 2 d x d t < ∞ , then all solution components decay in the sense that n ( · , t ) → 0 , c ( · , t ) → 0 and u ( · , t ) → 0 in L ∞ ( Ω ) as t → ∞ .