A further study on a 3D chemotaxis–Stokes system with tensor-valued sensitivity. (April 2019) - He, Pu; Wang, Yulan; Zhao, Li
Abstract: We consider the three-dimensional chemotaxis–Stokes system n t + u ⋅ ∇ n = Δ 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, 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. Here the tensor-valued sensitivity S fulfills | S ( x, n, c ) | ≤ C s ( 1 + n ) − α with some C s > 0 and α > 0 . It has been shown in Wang–Pang–Li (2016) that if α > 1 6, this system possesses a global bounded classical solution for any given initial data. We shall improve this result to the case of arbitrary α > 0, and thus solve the problem left in Wang–Cao (2015).