Global weak solutions for a 3D chemotaxis–Stokes system with slow [formula omitted]-Laplacian diffusion and rotation

Abstract

In this paper we study the chemotaxis–Stokes system with slow p-Laplacian diffusion and rotation: nt+u⋅∇n=∇⋅(|∇n|p−2∇n)−∇⋅(nS(x,n,c)⋅∇c), ct+u⋅∇c=Δc−nc, ut+∇P=Δu+n∇ϕ+f(x,t) and ∇⋅u=0 in a bounded domain Ω⊂R3 with p>2, subject to the Neumann–Neumann–Dirichlet boundary conditions, where ϕ:Ω̄→R, f:Ω̄×[0,∞)→R3 and S:Ω̄×[0,∞)2→R3×3 are given sufficiently smooth functions with f bounded in Ω×(0,∞), |S(x,n,c)|≤S0(c)(1+n)−α for (x,n,c)∈Ω̄×[0,∞)2 with α≥0, and nondecreasing function S0:[0,∞)→[0,∞). It is proved that the problem possesses a globally bounded weak solution provided α+43p>259 and 11p+6α+2αp>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.