Boundedness in chemotaxis–Stokes system with rotational flux term. (February 2019) - Zhou, Shuangshuang
Abstract: We consider the following chemotaxis–Stokessystem with rotation n t = Δ n − ∇ ⋅ ( n S ( x, n, c ) ⋅ ∇ c ) − u ⋅ ∇ n, c t = Δ c − f ( x, n, c ) − u ⋅ ∇ c, u t = Δ u + ∇ P + n ∇ ϕ, ∇ ⋅ u = 0 in Ω × ( 0, T ), subject to the non-flux boundary conditions for n and c, as well as the Dirichlet boundary condition for u, where the bounded smooth domain Ω ⊂ R 3, the matrix-valued function S ∈ C 2 ( Ω ̄ × [ 0, ∞ ) 2 ; R 3 × 3 ) fulfills | S ( x, n, c ) | ≤ S 0 ( c ) ( 1 + n ) θ for all ( x, n, c ) ∈ Ω ̄ × [ 0, ∞ ) 2 with S 0 nondecreasing, and f ∈ C 1 ( Ω ̄ × [ 0, ∞ ) 2 ; R ) satisfies 0 ≤ f ( x, n, c ) ≤ f 0 ( c ) ( n + 1 ) with f 0 nondecreasing and f ( x, n, 0 ) = 0 . It was proved that for any θ > 0, the initial–boundary value problem possesses a unique globally bounded classical solution.