2019/03/04 by Zheng, Jiashan
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1903.01033
This paper deals with a boundary-value problem for a coupled chemotaxis-Navier-Stokes system involving tensor-valued sensitivity with saturation \ nt+u⋅∇ n=Δn-∇⋅(nS(x,n,c)∇ c), x∈ Ω, t · gt;0, ct+u⋅∇ c=Δc-c+n, x∈ Ω, t · gt;0,
ut+κ(u ⋅ ∇)u+∇ P=Δu+n∇ ϕ, x∈ Ω, t · gt;0,
∇⋅ u=0, x∈ Ω, t · gt;0,. which describes chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells, where κ∈ ℝ,ϕ∈ W2,∞(Ω) and S is a given function with values in ℝ2×2 which fulfills |S(x,n,c)| ≤ CS (1 + n)-α with some C S > 0 and α≥ 0. If α>0 and Ω⊆ ℝ2 is a \bf bounded domain with smooth boundary, then for all reasonably regular initial data, a corresponding initial-boundary value problem for (KSNF) possesses a global classical solution which is bounded on Ω×(0,∞). This extends a recent result by Wang-Winkler-Xiang (Annali della Scuola Normale Superiore di Pisa-Classe di Scienze. XVIII, (2018), 2036--2145) which asserts global existence of bounded solutions under the constraint Ω⊆ ℝ2 is a bounded \bf convex domain with smooth boundary. Moreover, we shall improve the result of Wang-Xiang (J. Diff. Eqns., 259(2015), 7578--7609), who proved the possibility of global and bounded, in the case that \bfκ≡0 and α>0. In comparison to the result for the corresponding fluid-free system, the \bf optimal condition on the parameter α for both \bf global existence and \bf boundedness are obtained.