Eventual smoothness and asymptotic stabilization in a two-dimensional logarithmic chemotaxis-Navier–Stokes system with nutrient-supported proliferation and signal consumption

• Published in: Journal of Differential Equations 2024-01, Vol.378, p.171-203
• Main Authors: Wang, Yifu, Liu, Ji
• Format: Article
• Language: English
• Subjects: Chemotaxis; Eventual regularity; Generalized solution; Navier–Stokes; Stabilization
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2023.09.026

In this paper, we study the influence of nutrient-dependent cell proliferation on large time behavior of the solutions to an associated initial-boundary problem of(⋆){nt+u⋅∇n=Δn−∇⋅(nc∇c)+nc,x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+(u⋅∇)u=Δu+∇P+n∇Φ,x∈Ω,t>0, subject to no-flux/no-flux/Dirichlet boundary conditions in a smoothly bounded domain Ω⊂R2, where +nc in the first equation denotes cell reproduction in dependence on nutrients. It is proved that the initial-boundary problem is globally solvable in a generalized sense for any appropriately regular initial data, and that the generalized solutions thereof emanating from suitably small initial data will become smooth from a finite waiting time and exponentially approach (1|Ω|∫Ω(n0+c0),0,0) as t→∞. As compared to a precedent result obtained without cell proliferation, the complexity caused by +nc in the first equation requires some conditions on the initial data c0 rather only on n0 in order to derive the large time behavior of the solutions thereof, which seems comprehensible because of the necessity for offsetting the possibly increasing trend of cell by +nc.