On a three-dimensional quasilinear Keller–Segel–Stokes system with indirect signal production

• Published in: Archiv der Mathematik 2023, Vol.120 (1), p.77-87
• Main Author: Zheng, Pan
• Format: Article
• Language: English
• Subjects: Boundary conditions; Mathematics; Mathematics and Statistics
• ISSN: 0003-889X; 1420-8938
• DOI: 10.1007/s00013-022-01805-2

This paper is concerned with the effect of the indirect signal production mechanism on global boundedness of solutions for the following Keller–Segel–Stokes system n t + u · ∇ n = ∇ · ( D ( n ) ∇ n - S ( n ) ∇ c ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) , c t + u · ∇ c = Δ c - c + v , ( x , t ) ∈ Ω × ( 0 , ∞ ) , v t + u · ∇ v = Δ v - v + n , ( x , t ) ∈ Ω × ( 0 , ∞ ) , u t = Δ u + ∇ P + n ∇ ϕ , ( x , t ) ∈ Ω × ( 0 , ∞ ) , ∇ · u = 0 , ( x , t ) ∈ Ω × ( 0 , ∞ ) , in a smoothly bounded domain Ω ⊂ R 3 under zero-flux boundary conditions for n ,  c ,  v and no-slip boundary conditions for u , where D ( n ) and S ( n ) denote the nonlinear diffusion and sensitivity, respectively, u represents the velocity of the fluid, P is the pressure within the fluid, and ϕ is the gravitational potential. By means of the novel conditional estimates for ∇ c and u , it is proved that for all appropriately regular nonnegative initial data, this model has a globally bounded classical solution provided that the functions D , S ∈ C 2 ( [ 0 , ∞ ) ) satisfy D ( n ) ≥ K 1 ( n + 1 ) - m and S ( n ) ≤ K 2 n ( n + 1 ) α - 1 with K 1 , K 2 > 0 and α + m < 8 9 , which improves the previous subcritical exponent α + m < 2 3 in the direct signaling Keller–Segel–Stokes system by Winkler (Appl Math Lett 112:106785, 2021). It is shown that the indirect signal production mechanism can be beneficial to the global boundedness of solutions for the three-dimensional quasilinear Keller–Segel–Stokes system.