A new result for the global existence (and boundedness) and regularity of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization

Abstract

Abstract This paper deals with the following quasilinear Keller-Segel-Navier-Stokes system modeling coral fertilization (*) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n S ( x , n , c ) ∇ c ) − n m , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − c + m , x ∈ Ω , t > 0 , m t + u ⋅ ∇ m = Δ m − n m , x ∈ Ω , t > 0 , u t + κ ( u ⋅ ∇ ) u + ∇ P = Δ u + ( n + m ) ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 under no-flux boundary conditions in a bounded domain Ω ⊂ R 3 with smooth boundary, where ϕ ∈ W 2 , ∞ ( Ω ) . Here S ( x , n , c ) denotes the rotational effect which satisfies S ∈ C 2 ( Ω ¯ × [ 0 , ∞ ) 2 ; R 3 × 3 ) and | S ( x , n , c ) | ≤ S 0 ( c ) ( 1 + n ) − α with α ≥ 0 and some nonnegative nondecreasing function S 0 . Based on a new weighted estimate and some careful analysis, if α > 0 , then for any κ ∈ R , system ( ⁎ ) possesses a global weak solution. Furthermore, for any p > 1 , this solution is uniformly bounded with respect to the norm in L p ( Ω ) × L ∞ ( Ω ) × L ∞ ( Ω ) × L 2 ( Ω ; R 3 ) . Moreover, if κ = 0 , then system ( ⁎ ) admits a classical solution which is global in time and bounded.