Global weak solutions for a 3D chemotaxis–Stokes system with slow p-Laplacian diffusion and rotation. (December 2020) - Zhuang, Mengdi; Wang, Wei; Zheng, Sining
Abstract: In this paper we study the chemotaxis–Stokes system with slow p -Laplacian diffusion and rotation: n t + u ⋅ ∇ n = ∇ ⋅ ( | ∇ n | p − 2 ∇ n ) − ∇ ⋅ ( n S ( x, n, c ) ⋅ ∇ c ), c t + u ⋅ ∇ c = Δ c − n c, u t + ∇ P = Δ u + n ∇ ϕ + f ( x, t ) and ∇ ⋅ u = 0 in a bounded domain Ω ⊂ R 3 with p > 2, subject to the Neumann–Neumann–Dirichlet boundary conditions, where ϕ : Ω ̄ → R, f : Ω ̄ × [ 0, ∞ ) → R 3 and S : Ω ̄ × [ 0, ∞ ) 2 → R 3 × 3 are given sufficiently smooth functions with f bounded in Ω × ( 0, ∞ ), | S ( x, n, c ) | ≤ S 0 ( c ) ( 1 + n ) − α for ( x, n, c ) ∈ Ω ̄ × [ 0, ∞ ) 2 with α ≥ 0, and nondecreasing function S 0 : [ 0, ∞ ) → [ 0, ∞ ) . It is proved that the problem possesses a globally bounded weak solution provided α + 4 3 p > 25 9 and 11 p + 6 α + 2 α p > 23 . This extends the current global boundedness result by Tao and Li (2020), where the case of α = 0 was well solved. It is mentioned that, without constructing coupled energy functionals, the technique used in the present paper is somewhat different.