Influence of flux limitation on large time behavior in a three-dimensional chemotaxis-Stokes system modeling coral fertilization
In this paper, we consider the following system ⎩⎨⎧nt+u⋅∇nct+u⋅∇cmt+u⋅∇mut=Δn−∇⋅(nS(∣∇c∣2)∇c)−nm,=Δc−c+m,=Δm−mn,=Δu+∇P+(n+m)∇Φ,∇⋅u=0 which models the process of coral fertilization, in a smoothly three-dimensional bounded domain, where S is a given function fulfilling ∣S(σ)∣≤KS(1+σ)−2θ,σ≥0 with some KS>0. Based on conditional estimates of the quantity c and the gradients thereof, a relatively compressed argument as compared to that proceeding in related precedents shows that if θ>0, then for any initial data with proper regularity an associated initial-boundary problem under no-flux/no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded, and which also enjoys the stabilization features in the sense that ∥n(⋅,t)−n∞∥L∞(Ω)+∥c(⋅,t)−m∞∥W1,∞(Ω)+∥m(⋅,t)−m∞∥W1,∞(Ω)+∥u(⋅,t)∥L∞(Ω)→0as t→∞ with n∞:=∣Ω∣1{∫Ωn0−∫Ωm0}+ and m∞:=∣Ω∣1{∫Ωm0−∫Ωn0}+.