LAUSR

Abstract In this paper, we discuss the following chemotaxis-Stokes system with nonlinear diffusion and rotation { n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 with m > 0 , where Ω is a bounded domain in R 3 , S ( x , n , c ) is a chemotactic sensitivity tensor satisfying S ∈ C 2 ( Ω ¯ × [ 0 , ∞ ) 2 ; R 3 × 3 ) and | S ( x , n , c ) | ≤ ( 1 + n ) − α S 0 ( c ) for all ( x , n , c ) ∈ Ω × [ 0 , ∞ ) 2 with α ≥ 0 and some nondecreasing function S 0 : [ 0 , ∞ ) → [ 0 , ∞ ) . We can show that if m + α > 10 9 , then for any reasonably smooth initial data, the corresponding initial-boundary problem possesses a globally bounded weak solution. The result extends the previous global boundedness result for m + α > 10 9 , α > 0 and m + 5 4 α > 9 8 [15] , and m > 9 8 and α = 0 [24] . Without using the conventional free-energy inequality see [15] , [24] , we use a new iterative bootstrap procedure to estimate the bounds on ∫ Ω n e p for p > 3 2 , which is a crucial step to obtain our main result.