Global Bounded Solution in a Chemotaxis-Stokes Model with Porous Medium Diffusion and Singular Sensitivity.

This article is concerned with a chemotaxis-Stokes system with porous medium diffusion and singular sensitivity: { n t + u ⋅ ∇ n = ∇ ⋅ (D (n) ∇ n) − ∇ ⋅ (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 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 <graphic href="10440_2023_599_Article_Equa.gif"></graphic> in a bounded domain Ω ⊂ R N with 2 ≤ N ≤ 3 , where D ∈ C 0 ([ 0 , ∞)) ∩ C 2 ((0 , ∞)) and S ∈ C 2 (Ω ¯ × [ 0 , ∞) 2 ; R N × N) . The global solvability of the system in a natural weak sense is obtained under the conditions that D (n) ≥ k D n m − 1 and | S (x , n , c) | ≤ S 0 (c) c α for all (x , n , c) ∈ Ω × (0 , ∞) 2 with some k D > 0 , m > 3 N − 2 2 N , α ∈ [ 0 , 1) and some nondecreasing S 0 : (0 , ∞) → (0 , ∞) . Moreover, in the case that m = 3 N − 2 2 N and α ∈ [ 0 , 1) , we also get the global weak solutions under smallness assumptions on the initial data ∥ n 0 ∥ L 1 (Ω) and ∥ c 0 ∥ L ∞ (Ω) . [ABSTRACT FROM AUTHOR]